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| The '''pair distribution function''' (PDF) describes the distribution of distances between pairs of particles contained within a given volume. Mathematically, if ''a'' and ''b'' are two particles in a fluid, the PDF of ''b'' with respect to ''a'', denoted by <math>g_{ab}(\vec{r})</math> is the probability of finding the particle ''b'' at the distance <math>\vec{r}</math> from ''a'', with ''a'' taken as the origin of coordinates. | | Greetings. The writer's title is Phebe and she feels comfy when people use the complete title. For many years he's been operating as a meter reader and it's some thing he truly enjoy. For a while she's been in South Dakota. One of the very best issues in the world for him is to collect badges but he is struggling to find time for it.<br><br>My website ... [http://calvaryhill.net/xe/board_DsjO50/23648 home std test] |
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| == Overview ==
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| The pair distribution function is used to describe the distribution of objects within a medium (for example, oranges in a crate or nitrogen molecules in a gas cylinder). If the medium is homogeneous (i.e. every spatial location has identical properties), then there is an equal probability density for finding an object at any position <math>\vec{r}</math>:
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| :<math>p(\vec{r})=1/V</math>,
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| where <math>V</math> is the volume of the container. On the other hand, the likelihood of finding ''pairs of objects'' at given positions (i.e. the two-body probability density) is not uniform. For example, pairs of hard balls must be separated by at least the diameter of a ball. The pair distribution function <math>g(\vec{r},\vec{r'})</math> is obtained by scaling the two-body probability density function by the total number of objects <math>N</math> and the size of the container:
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| :<math>g(\vec{r}, \vec{r}') = p(\vec{r},\vec{r}') V^2 \frac{N-1}{N}</math>.
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| In the common case where the number of objects in the container is large, this simplifies to give:
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| :<math>g(\vec{r}, \vec{r}') \approx p(\vec{r},\vec{r}') V^2</math>.
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| == Simple models and general properties ==
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| The simplest possible pair distribution function assumes that all object locations are mutually independent, giving:
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| :<math>g(\vec{r})=1</math>,
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| where <math>\vec{r}</math> is the separation between a pair of objects. However, this is inaccurate in the case of hard objects as discussed above, because it does not account for the minimum separation required between objects. The hole-correction (HC) approximation provides a better model:
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| :<math>g(r) =
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| \begin{cases}
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| 0,&r<b,\\
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| 1,&r\geq{}b
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| \end{cases},
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| </math> | |
| where <math>b</math> is the diameter of one of the objects.
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| Although the HC approximation gives a reasonable description of sparsely packed objects, it breaks down for dense packing. This may be illustrated by considering a box completely filled by identical hard balls so that each ball touches its neighbours. In this case, every pair of balls in the box is separated by a distance of exactly <math>r=nb</math> where <math>n</math> is a positive whole number. The pair distribution for a volume completely filled by hard spheres is therefore a set of [[Dirac delta function]]s of the form:
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| :<math>g(r)=\sum\limits_i\delta(r-ib)</math>.
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| Finally, it may be noted that a pair of objects which are separated by a large distance have no influence on each other's position (provided that the container is not completely filled). Therefore,
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| :<math>\lim\limits_{r\to\infty}g(r) = 1</math>.
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| In general, a pair distribution function will take a form somewhere between the sparsely packed (HC approximation) and the densely packed (delta function) models, depending on the packing density <math>f</math>.
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| == Radial pair distributions ==
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| Of special practical importance is the radial pair distribution function, which is independent of orientation. It is a major descriptor for the atomic structure of amorphous materials (glasses, polymers) and liquids. The radial PDF can be calculated directly from physical measurements like [[light scattering]] or [[x-ray powder diffraction]] through the use of [[Fourier Transform]].
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| In Statistical Mechanics the PDF is given by the expression:
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| <math>
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| g_{ab}(r) = \frac{1}{N_{a} N_b}\sum\limits_{i=1}^{N_a} \sum\limits_{j=1}^{N_b} \langle \delta( \vert \mathbf{r}_{ij} \vert -r)\rangle
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| </math>
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| The [http://www.diffpy.org Diffpy] project is used to match crystal structures with PDF data derived from X-ray or neutron diffraction data.
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| == See also ==
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| * [[Radial distribution function]]
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| * [[classical-map hypernetted-chain method]]
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| [[Category:Statistical mechanics]]
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| [[Category:Condensed matter physics]]
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