Content deleted Content added
Citation bot (talk | contribs) Add: s2cid, pmid, page, author pars. 1-1. Removed URL that duplicated unique identifier. Removed parameters. Some additions/deletions were actually parameter name changes. | You can use this bot yourself. Report bugs here. | Suggested by SemperIocundus | via #UCB_webform |
|||
Line 121:
{{NumBlk|:| <math>g(r) = \exp \left [ -\frac{u(r)}{kT} \right ] y(r) \quad \mathrm{with} \quad y(r) = 1 + \sum_{n=1}^{\infty} \rho ^n y_n (r)</math>.|{{EquationRef|12}}}}
This similarity is not accidental; indeed, substituting ({{EquationNote|12}}) in the relations above for the thermodynamic parameters (Equations {{EquationNote|7}}, {{EquationNote|9}} and {{EquationNote|10}}) yields the corresponding virial expansions.<ref name="Barker:1976">{{Cite journal | last1 = Barker | first1 = J. | last2 = Henderson | first2 = D. | doi = 10.1103/RevModPhys.48.587 | title = What is "liquid"? Understanding the states of matter | journal = Reviews of Modern Physics | volume = 48 | issue = 4 | pages = 587 | year = 1976 | pmid = | pmc = |bibcode = 1976RvMP...48..587B }}</ref> The auxiliary function <math>y(r)</math> is known as the ''cavity distribution function''.<ref name="HansenMcDonald2005" />{{rp|Table 4.1}} It has been shown that for classical fluids at a fixed density and a fixed positive temperature, the effective pair potential that generates a given <math>g(r)</math> under equilibrium is unique up to an additive constant, if it exists.<ref>{{Cite journal|last=Henderson|first=R. L.|date=1974-09-09|title=A uniqueness theorem for fluid pair correlation functions
In recent years, some attention has been given to develop Pair Correlation Functions for spatially-discrete data such as lattices or networks.<ref>{{cite journal |last1=Gavagnin |first1=Enrico |title=Pair correlation functions for identifying spatial correlation in discrete domains |journal=Physical Review E |date=4 June 2018 |volume=97 |issue=1 |page=062104 |doi=10.1103/PhysRevE.97.062104|pmid=30011502 |arxiv=1804.03452 |s2cid=50780864 }}</ref>
==Experimental==
Line 129:
One can determine <math>g(r)</math> indirectly (via its relation with the structure factor <math>S(q)</math>) using [[neutron scattering]] or [[x-ray scattering]] data. The technique can be used at very short length scales (down to the atomic level<ref>{{Cite journal | last1 = Yarnell | first1 = J. | last2 = Katz | first2 = M. | last3 = Wenzel | first3 = R. | last4 = Koenig | first4 = S. | title = Structure Factor and Radial Distribution Function for Liquid Argon at 85 °K | doi = 10.1103/PhysRevA.7.2130 | journal = Physical Review A | volume = 7 | issue = 6 | pages = 2130 | year = 1973 | pmid = | pmc = |bibcode = 1973PhRvA...7.2130Y }}</ref>) but involves significant space and time averaging (over the sample size and the acquisition time, respectively). In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals<ref>{{Cite journal | last1 = Gingrich | first1 = N. S. | last2 = Heaton | first2 = L. | doi = 10.1063/1.1731688 | title = Structure of Alkali Metals in the Liquid State | journal = The Journal of Chemical Physics | volume = 34 | issue = 3 | pages = 873 | year = 1961 | pmid = | pmc = |bibcode = 1961JChPh..34..873G }}</ref> to charged colloids.<ref>{{Cite journal | last1 = Sirota | first1 = E. | last2 = Ou-Yang | first2 = H. | last3 = Sinha | first3 = S. | last4 = Chaikin | first4 = P. | last5 = Axe | first5 = J. | last6 = Fujii | first6 = Y. | doi = 10.1103/PhysRevLett.62.1524 | title = Complete phase diagram of a charged colloidal system: A synchro- tron x-ray scattering study | journal = Physical Review Letters | volume = 62 | issue = 13 | pages = 1524–1527 | year = 1989 | pmid = 10039696| pmc = |bibcode = 1989PhRvL..62.1524S }}</ref> Going from the experimental <math>S(q)</math> to <math>g(r)</math> is not straightforward and the analysis can be quite involved.<ref>{{Cite journal | last1 = Pedersen | first1 = J. S. | doi = 10.1016/S0001-8686(97)00312-6 | title = Analysis of small-angle scattering data from colloids and polymer solutions: Modeling and least-squares fitting | journal = Advances in Colloid and Interface Science | volume = 70 | pages = 171–201 | year = 1997 | pmid = | pmc = }}</ref>
It is also possible to calculate <math>g(r)</math> directly by extracting particle positions from traditional or confocal microscopy.<ref>{{Cite journal | last1 = Crocker | first1 = J. C.| last2 = Grier | first2 = D. G.| doi = 10.1006/jcis.1996.0217 | title = Methods of Digital Video Microscopy for Colloidal Studies | journal = Journal of Colloid and Interface Science | volume = 179 | issue = 1| pages = 298–310| year = 1996 | pmid = | pmc = | bibcode = 1996JCIS..179..298C}}</ref> This technique is limited to particles large enough for optical detection (in the micrometer range), but it has the advantage of being time-resolved so that, aside from the statical information, it also gives access to dynamical parameters (e.g. [[diffusion constant]]s<ref>{{Cite journal | last1 = Nakroshis | first1 = P. | last2 = Amoroso | first2 = M. | last3 = Legere | first3 = J. | last4 = Smith | first4 = C. | title = Measuring Boltzmann's constant using video microscopy of Brownian motion | doi = 10.1119/1.1542619 | journal = American Journal of Physics | volume = 71 | issue = 6 | pages = 568 | year = 2003 | pmid = | pmc = |bibcode = 2003AmJPh..71..568N }}</ref>) and also space-resolved (to the level of the individual particle), allowing it to reveal the morphology and dynamics of local structures in colloidal crystals,<ref>{{Cite journal | last1 = Gasser | first1 = U. | last2 = Weeks | first2 = E. R. | last3 = Schofield | first3 = A. | last4 = Pusey | first4 = P. N. | last5 = Weitz | first5 = D. A. | title = Real-Space Imaging of Nucleation and Growth in Colloidal Crystallization | doi = 10.1126/science.1058457 | journal = Science | volume = 292 | issue = 5515 | pages = 258–262 | year = 2001 | pmid = 11303095| pmc = |bibcode = 2001Sci...292..258G | s2cid = 6590089 }}</ref> glasses,<ref>M.I. Ojovan, D.V. Louzguine-Luzgin. Revealing Structural Changes at Glass Transition via Radial Distribution Functions. J. Phys. Chem. B, 124 (15), 3186-3194 (2020) https://doi.org/10.1021/acs.jpcb.0c00214</ref>,<ref>{{Cite journal | last1 = Weeks | first1 = E. R. | last2 = Crocker | first2 = J. C. | last3 = Levitt | first3 = A. C. | last4 = Schofield | first4 = A. | last5 = Weitz | first5 = D. A. | title = Three-Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition | doi = 10.1126/science.287.5453.627 | journal = Science | volume = 287 | issue = 5453 | pages = 627–631 | year = 2000 | pmid = 10649991| pmc = |bibcode = 2000Sci...287..627W }}</ref> gels,<ref>{{Cite journal | last1 = Cipelletti | first1 = L. | last2 = Manley | first2 = S. | last3 = Ball | first3 = R. C. | last4 = Weitz | first4 = D. A. | title = Universal Aging Features in the Restructuring of Fractal Colloidal Gels | doi = 10.1103/PhysRevLett.84.2275 | journal = Physical Review Letters | volume = 84 | issue = 10 | pages = 2275–2278 | year = 2000 | pmid = 11017262| pmc = |bibcode = 2000PhRvL..84.2275C }}</ref><ref>{{Cite journal | last1 = Varadan | first1 = P. | last2 = Solomon | first2 = M. J. | doi = 10.1021/la026303j | title = Direct Visualization of Long-Range Heterogeneous Structure in Dense Colloidal Gels | journal = Langmuir | volume = 19 | issue = 3 | pages = 509 | year = 2003 | pmid = | pmc = }}</ref> and hydrodynamic interactions.<ref>{{Cite journal | last1 = Gao | first1 = C. | last2 = Kulkarni | first2 = S. D. | last3 = Morris | first3 = J. F. | last4 = Gilchrist | first4 = J. F. | title = Direct investigation of anisotropic suspension structure in pressure-driven flow | doi = 10.1103/PhysRevE.81.041403 | journal = Physical Review E | volume = 81 | issue = 4 | pages = 041403 | year = 2010 | pmid = 20481723| pmc = |bibcode = 2010PhRvE..81d1403G }}</ref>
Direct visualization of a full (distance-dependent and angle-dependent) pair correlation function was achieved by a [[scanning tunneling microscope|scanning tunneling microscopy]] in the case of 2D molecular gases.<ref>{{Cite journal|
==Higher-order correlation functions==
|