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| | I'm Lisa and I live in a seaside city in northern Italy, Calceranica Al Lago. I'm 29 and I'm will soon finish my study at Athletics and Physical Education.<br><br>My web-site; [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine handbags] |
| '''Mean inter-particle distance''' (or mean inter-particle separation) is the mean distance between microscopic particles (usually [[atoms]] or [[molecules]]) in a macroscopic body. | |
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| ==Ambiguity==
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| From the very general considerations, the mean inter-particle distance is proportional to the size of the per-particle volume <math>1/n</math>, i.e.,
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| : <math>\langle r \rangle \sim 1/n^{1/3},</math>
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| where <math>n = N/V</math> is the [[particle density]]. However, barring a few simple cases such as the [[ideal gas]] model, precise calculations of the proportionality factor are impossible analytically. Therefore, approximate expressions are often used. One such an estimation is the [[Wigner-Seitz radius]]
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| : <math>\left( \frac{3}{4 \pi n} \right)^{1/3},</math>
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| which corresponds to the radius of a sphere having per-particle volume <math>1/n</math>. Another popular definition is
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| : <math>1/n^{1/3}</math>,
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| corresponding to the length of the edge of the cube with the per-particle volume <math>1/n</math>. The two definitions differ by a factor of approximately <math>1.61</math>, so one has to exercise care if an article fails to define the parameter exactly. On the other hand, it is often used in qualitative statements where such a numeric factor is either irrelevant or plays an insignificant role, e.g.,
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| * "a potential energy ... is proportional to some power n of the inter-particle distance r" ([[Virial theorem]])
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| * "the inter-particle distance is much larger than the thermal de Broglie wavelength" ([[Kinetic theory]])
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| ==Ideal gas==
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| ===Nearest neighbor distribution===
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| [[File:PDF NN in ideal gas.svg|thumb|300px|PDF of the NN distances in an ideal gas.]]
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| We want to calculate [[probability distribution function]] of distance to the nearest neighbor (NN) particle. (The problem was first considered by [[Paul Hertz]];<ref>{{Cite journal
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| | doi = 10.1007/BF01450410
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| | issn = 0025-5831
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| | volume = 67
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| | issue = 3
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| | pages = 387–398
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| | last = Hertz
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| | first = Paul
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| | title = Über den gegenseitigen durchschnittlichen Abstand von Punkten, die mit bekannter mittlerer Dichte im Raume angeordnet sind
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| | journal = Mathematische Annalen
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| | accessdate = 2011-03-03
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| | year = 1909
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| | url = http://www.springerlink.com/content/q133104qq7596l37/
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| }}</ref> for a modern derivation see, e.g.,.<ref>{{Cite journal
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| | doi = 10.1103/RevModPhys.15.1
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| | volume = 15
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| | issue = 1
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| | pages = 1
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| | last = Chandrasekhar
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| | first = S.
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| | title = Stochastic Problems in Physics and Astronomy
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| | journal = Reviews of Modern Physics
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| | accessdate = 2011-03-01
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| | date = 1943-01-01
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| | url = http://link.aps.org/doi/10.1103/RevModPhys.15.1
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| | bibcode=1943RvMP...15....1C
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| }}</ref>) Let us assume <math>N</math> particles inside a sphere having volume <math>V</math>, so that <math>n = N/V</math>. Note that since the particles in the ideal gas are non-interacting, the probability to find a particle at a certain distance from another particle is the same as probability to find a particle at the same distance from any other point; we shall use the center of the sphere.
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| An NN particle at distance <math>r</math> means exactly one of the <math>N</math> particles resides at that distance while the rest
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| <math>N - 1</math> particles are at larger distances, i.e., they are somewhere outside the sphere with radius <math>r</math>.
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| The probability to find a particle at the distance from the origin between <math>r</math> and <math>r + dr</math> is
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| <math>(4 \pi r^2 N/V) dr</math>, while the probability to find a particle outside that sphere is <math>1 - 4\pi r^3/3V</math>. The sought-for expression is then
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| :<math>P_N(r)dr = 4 \pi r^2 dr\frac{N}{V}\left(1 - \frac{4\pi}{3}r^3/V \right)^{N - 1} =
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| \frac{3}{a}\left(\frac{r}{a}\right)^2 dr \left(1 - \left(\frac{r}{a}\right)^3 \frac{1}{N} \right)^{N - 1}\,,</math>
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| where we substituted
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| : <math>a = \left( \frac{3}{4 \pi n} \right)^{1/3}.</math>
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| Finally, taking the <math>N \rightarrow \infty</math> limit and using <math>\lim_{x \rightarrow \infty}\left(1 + \frac{1}{x}\right)^x = e</math>, we obtain
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| :<math>P(r) = \frac{3}{a}\left(\frac{r}{a}\right)^2 e^{-(r/a)^3}\,.</math>
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| One can immediately check that
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| :<math>\int_{0}^{\infty}P(r)dr = 1\,.</math>
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| The distribution peaks at
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| :<math>r_{\text{peak}} = \left(2/3\right)^{1/3} a \approx 0.874 a\,.</math>
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| ===Mean distance and higher NN distribution moments===
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| :<math> \langle r^k \rangle = \int_{0}^{\infty}P(r) r^k dr = 3 a^k\int_{0}^{\infty}x^{k+2}e^{-x^3}dx\,,</math>
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| or, using the <math>t = x^3</math> substitution,
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| :<math> \langle r^k \rangle = a^k \int_{0}^{\infty}t^{k/3}e^{-t}dt = a^k \Gamma(1 + \frac{k}{3})\,,</math>
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| where <math>\Gamma</math> is the [[gamma function]]. Thus,
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| :<math> \langle r^k \rangle = a^k \Gamma(1 + \frac{k}{3})\,.</math>
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| In particular,
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| :<math> \langle r \rangle = a \Gamma(\frac{4}{3}) = \frac{a}{3} \Gamma(\frac{1}{3}) \approx 0.893 a\,.</math>
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| ==References==
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| <references />
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| ==See also==
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| * [[Wigner-Seitz radius]]
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| [[Category:Concepts in physics]]
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| [[Category:Density]]
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I'm Lisa and I live in a seaside city in northern Italy, Calceranica Al Lago. I'm 29 and I'm will soon finish my study at Athletics and Physical Education.
My web-site; Celine handbags