Radial distribution function: Difference between revisions

Revision as of 12:49, 6 November 2013

Revision as of 15:29, 18 December 2013

Line 10:

The RDF is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The histogram is then normalized with respect to an ideal gas, where particle histograms are completely uncorrelated. For three dimensions, this normalization is the number density of the system multiplied by the volume of the spherical shell, which mathematically can be expressed as <math>g(r)_I = 4\pi r^2\rho dr</math>, where <math>\rho</math> is the number density.

Given a [[potential energy]] function, the radial distribution function can be computed either via computer simulation methods like the [[Monte Carlo method]], or via the [[Ornstein-Zernike equation]], using approximative closure relations like the [[Percus-Yevick approximation]] or the [[Hypernetted-chain equation|Hypernetted Chain Theory]]. It can also be determined experimentally, by radiation scattering techniques or by direct visualization for large enough (micronmicrometer-sized) particles via traditional or confocal microscopy.

The radial distribution function is of fundamental importance in thermodynamics because the macroscopic thermodynamic quantities can usually be determined from <math>g(r)</math>.

Line 48:

===The structure factor===

The second-order correlation function <math>g^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2})</math> is of special importance, as it is directly related (via a [[Fourier transform]]) to the [[structure factor]] of the system and can thus be determined experimentally using [[X-ray diffraction]] or [[neutron diffraction]].<ref>{{citation needed</ref>|date=December 2013}}

If the system consists of spherically symmetric particles, <math>g^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2})</math> depends only on the relative distance between them, <math>\mathbf{r}_{12} = \mathbf{r}_{2} - \mathbf{r}_{1} </math>. We will drop the sub- and superscript: <math>\textstyle g(\mathbf{r})\equiv g^{(2)}(\mathbf{r}_{12})</math>. Taking particle 0 as fixed at the origin of the coordinates, <math>\textstyle \rho g(\mathbf{r}) \mathrm{d}\rm{r} = \mathrm{d} n (\mathbf{r})</math> is the number of particles (among the remaining <math>N-1</math>) to be found in the volume <math>\textstyle \mathrm{d}\mathbf{r}</math> around the position <math>\textstyle \mathbf{r}</math>.

Line 75:

===The potential of mean force===

It can be shown<ref name="Chandler1987">See{{cite book |author=[[David Chandler (chemist)|Chandler, sectionD.]] |year=1987 |title=Introduction to Modern Statistical Mechanics |publisher=Oxford University Press |section=7.3.}}</ref> that the radial distribution function is related to the two-particle [[potential of mean force]] <math>w^{(2)}(r)</math> by:

{{NumBlk|:| <math> g(r) = \exp \left [ -\frac{w^{(2)}(r)}{kT} \right ] </math>.|{{EquationRef|8}}}}

===The energy equation===

If the particles interact via identical pairwise potentials: <math>\textstyle U_{N} = \sum_{i > j = 1}^N u(\left | \mathbf{r}_i - \mathbf{r}_j \right |)</math>, the average internal energy per particle is:<ref name="HansenMcDonald2005">See{{cite book |author=[[Jean-Pierre Hansen|Hansen, &J. P.]] and McDonald, sectionI. 2.5R. |year=2005 |title=Theory of Simple Liquids |edition=3rd |publisher=Academic Press}}</ref>{{rp|Section 2.5}}

{{NumBlk|:| <math>\frac{\left \langle E \right \rangle}{N} = \frac{3}{2} kT + \frac{\left \langle U_{N} \right \rangle}{N} = \frac{3}{2} kT + \frac{\rho}{2}\int_V \mathrm{d} \mathbf{r} \, u(r)g(r, \rho, T) </math>.|{{EquationRef|9}}}}

Line 113:

{{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 doi|10.1103/RevModPhys.48.587}}</ref> The auxiliary function <math>y(r)</math> is known as the ''cavity distribution function''.<ref>See Hansen & McDonald,name="HansenMcDonald2005" />{{rp|Table 4.1 for a list of definitions.</ref>}}

==Experimental==

Line 119:

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 doi|10.1103/PhysRevA.7.2130}}</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 doi|10.1063/1.1731688}}</ref> to charged colloids.<ref>{{Cite doi|10.1103/PhysRevLett.62.1524}}</ref> It should be noted that 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 doi|10.1016/S0001-8686(97)00312-6}}</ref>

It is also possible to calculate <math>g(r)</math> directly by extracting particle positions from traditional or confocal microscopy.<ref>{{Cite doi|10.1006/jcis.1996.0217}}</ref> This technique is limited to particles large enough for optical detection (in the micronmicrometer 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 doi|10.1119/1.1542619}}</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 doi|10.1126/science.1058457}}</ref> glasses,<ref>{{Cite doi|10.1126/science.287.5453.627}}</ref> gels,<ref>{{Cite doi|10.1103/PhysRevLett.84.2275}}</ref><ref>{{Cite doi|10.1021/la026303j}}</ref> and hydrodynamic interactions.<ref>{{Cite doi|10.1103/PhysRevE.81.041403}}</ref>

==Higher-order correlation functions==

Line 126:

==References==

#[[David Chandler (chemist)|Chandler, D.]] (1987). Introduction to Modern Statistical Mechanics. Oxford University Press.

#[[Jean-Pierre Hansen|Hansen, J. P.]] and McDonald, I. R. (2005). Theory of Simple Liquids (3rd edition). Academic Press.

#Widom, B. (2002). Statistical Mechanics: A Concise Introduction for Chemists. Cambridge University Press.

#McQuarrie, D. A. (1976). Statistical Mechanics. Harper Collins Publishers.

{{Reflist}}

#* Widom, B. (2002). Statistical Mechanics: A Concise Introduction for Chemists. Cambridge University Press.

#* McQuarrie, D. A. (1976). Statistical Mechanics. Harper Collins Publishers.

==See also==