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Mean inter-particle distance

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Physical quantity

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Mean inter-particle distance (or mean inter-particle separation) is the mean distance between microscopic particles (usually atoms or molecules) in a macroscopic body.

Ambiguity

From the very general considerations, the mean inter-particle distance is proportional to the size of the per-particle volume 1 / n {\displaystyle 1/n} , i.e.,

⟨ r ⟩ ∼ 1 / n 1 / 3 , {\displaystyle \langle r\rangle \sim 1/n^{1/3},}

where n = N / V {\displaystyle n=N/V} 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 estimation is the Wigner–Seitz radius

( 3 4 π n ) 1 / 3 , {\displaystyle \left({\frac {3}{4\pi n}}\right)^{1/3},}

which corresponds to the radius of a sphere having per-particle volume 1 / n {\displaystyle 1/n} . Another popular definition is

1 / n 1 / 3 {\displaystyle 1/n^{1/3}} ,

corresponding to the length of the edge of the cube with the per-particle volume 1 / n {\displaystyle 1/n} . The two definitions differ by a factor of approximately 1.61 {\displaystyle 1.61} , 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.,

Ideal gas

Nearest neighbor distribution

PDF of the NN distances in an ideal gas.

We want to calculate probability distribution function of distance to the nearest neighbor (NN) particle. (The problem was first considered by Paul Hertz; for a modern derivation see, e.g.,.) Let us assume N {\displaystyle N} particles inside a sphere having volume V {\displaystyle V} , so that n = N / V {\displaystyle n=N/V} . Note that since the particles in the ideal gas are non-interacting, the probability of finding a particle at a certain distance from another particle is the same as the probability of finding a particle at the same distance from any other point; we shall use the center of the sphere.

An NN particle at a distance r {\displaystyle r} means exactly one of the N {\displaystyle N} particles resides at that distance while the rest N − 1 {\displaystyle N-1} particles are at larger distances, i.e., they are somewhere outside the sphere with radius r {\displaystyle r} .

The probability to find a particle at the distance from the origin between r {\displaystyle r} and r + d r {\displaystyle r+dr} is ( 4 π r 2 / V ) d r {\displaystyle (4\pi r^{2}/V)dr} , plus we have N {\displaystyle N} kinds of way to choose which particle, while the probability to find a particle outside that sphere is 1 − 4 π r 3 / 3 V {\displaystyle 1-4\pi r^{3}/3V} . The sought-for expression is then

P N ( r ) d r = 4 π r 2 d r N V ( 1 − 4 π 3 r 3 / V ) N − 1 = 3 a ( r a ) 2 d r ( 1 − ( r a ) 3 1 N ) N − 1 {\displaystyle P_{N}(r)dr=4\pi r^{2}dr{\frac {N}{V}}\left(1-{\frac {4\pi }{3}}r^{3}/V\right)^{N-1}={\frac {3}{a}}\left({\frac {r}{a}}\right)^{2}dr\left(1-\left({\frac {r}{a}}\right)^{3}{\frac {1}{N}}\right)^{N-1}\,}

where we substituted

1 V = 3 4 π N a 3 . {\displaystyle {\frac {1}{V}}={\frac {3}{4\pi Na^{3}}}.}

Note that a {\displaystyle a} is the Wigner-Seitz radius. Finally, taking the N → ∞ {\displaystyle N\rightarrow \infty } limit and using lim x → ∞ ( 1 + 1 x ) x = e {\displaystyle \lim _{x\rightarrow \infty }\left(1+{\frac {1}{x}}\right)^{x}=e} , we obtain

P ( r ) = 3 a ( r a ) 2 e − ( r / a ) 3 . {\displaystyle P(r)={\frac {3}{a}}\left({\frac {r}{a}}\right)^{2}e^{-(r/a)^{3}}\,.}

One can immediately check that

∫ 0 ∞ P ( r ) d r = 1 . {\displaystyle \int _{0}^{\infty }P(r)dr=1\,.}

The distribution peaks at

r peak = ( 2 / 3 ) 1 / 3 a ≈ 0.874 a . {\displaystyle r_{\text{peak}}=\left(2/3\right)^{1/3}a\approx 0.874a\,.}

Mean distance and higher moments

⟨ r k ⟩ = ∫ 0 ∞ P ( r ) r k d r = 3 a k ∫ 0 ∞ x k + 2 e − x 3 d x , {\displaystyle \langle r^{k}\rangle =\int _{0}^{\infty }P(r)r^{k}dr=3a^{k}\int _{0}^{\infty }x^{k+2}e^{-x^{3}}dx\,,}

or, using the t = x 3 {\displaystyle t=x^{3}} substitution,

⟨ r k ⟩ = a k ∫ 0 ∞ t k / 3 e − t d t = a k Γ ( 1 + k 3 ) , {\displaystyle \langle r^{k}\rangle =a^{k}\int _{0}^{\infty }t^{k/3}e^{-t}dt=a^{k}\Gamma (1+{\frac {k}{3}})\,,}

where Γ {\displaystyle \Gamma } is the gamma function. Thus,

⟨ r k ⟩ = a k Γ ( 1 + k 3 ) . {\displaystyle \langle r^{k}\rangle =a^{k}\Gamma (1+{\frac {k}{3}})\,.}

In particular,

⟨ r ⟩ = a Γ ( 4 3 ) = a 3 Γ ( 1 3 ) ≈ 0.893 a . {\displaystyle \langle r\rangle =a\Gamma ({\frac {4}{3}})={\frac {a}{3}}\Gamma ({\frac {1}{3}})\approx 0.893a\,.}

References

  1. Hertz, Paul (1909). "Über den gegenseitigen durchschnittlichen Abstand von Punkten, die mit bekannter mittlerer Dichte im Raume angeordnet sind". Mathematische Annalen. 67 (3): 387–398. doi:10.1007/BF01450410. ISSN 0025-5831. S2CID 120573104.
  2. Chandrasekhar, S. (1943-01-01). "Stochastic Problems in Physics and Astronomy". Reviews of Modern Physics. 15 (1): 1–89. Bibcode:1943RvMP...15....1C. doi:10.1103/RevModPhys.15.1.

See also

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