1/19 2/19 Structural quantities s08/3 Structure of fluids – correlation functions s08/3 Correlation functions: radial distribution function (also pair correlation/distribution function) g(r) = probability of finding a particle at distance r (from another particle), nor- malized to ideal gas structure factor (diffraction Fourier transform) → angular correlation function – good for small nonspherical molecules time autocorrelation functions randomly distributed molecules liquid (ideal gas) g(r) = pair correlation function = radial distribution function = probability density of finding a particle r apart from another particle, normalized so that for randomly distributed particles (ideal gas) it is 1 3/19 4/19 Structure of fluids – correlation functions s08/3 How to obtain structure – experiment s08/3 Diffraction (neutrons, X-ray, electrons) “structure factor” ⇒ inverse Fourier transform RDF ⇒ 5/19 6/19 How to obtain structure s08/3 Argon, hard spheres, water s08/3 4 The structure of simple fluid (argon, 3 HS) is organized by shells. The structure of water is determined 2 gHS(r) by the tetrahedral geometry of hy- 1 drogen bonds. After several molecular diameters, 0 the correlations decay to zero. 0 1 2 3 r 7/19 8/19 Running coordination number s08/3 RDF from simulations – simple s08/3 Also “cumulative radial distribution function” ρ N/V = number density Histogram of the number of particle pairs, , so that = N r 2 r [r Δr/2, r + Δr/2) alternatively = [r, r + Δr) N(r) = 4πρ g(r0)r0 dr0 ∈ − I Z0 The volume of the shell For rmin = first minimum on the RDF curve, N(rmin) = “coordination number” = 4π Δr 3 Δr 3 averaged number of molecules in the first shell ΔV = r + r 3 2 2 6 − − 5 Mean number of molecules around a selected particle in case of uniformly dis- tributed molecules (ideal gas, ρ = N/V): water 4 gOO(r) 1 NOO(r) ρΔV 2 3 gHO(r) NHO(r) (H around O) Sum over all particles (1/2 to count each pair just once): g(r) or N(r) 2 id. gas N ρΔV N = 1 2 Radial distribution function: 0 2 0 1 2 3 4 5 6 g r N N ( ) = 〈id. gas〉 = 〈 〉 r/A NρΔV N 9/19 10/19 Radial distribution function in the NVT ensemble s08/3 Calculation of RDF in simulations – exact + s08/3 3D (e.g., in periodic b.c.): Histogram of the count of pairs of particles, , so that r N ∈ I N N 1 ( ) = [r Δr/2, r + Δr/2), optionally = [r, r + Δr) g(r) g(r12) = − ... exp[ βU(r~1, r~2,..., r~N)] dr~3 ... dr~N I I ρ2Q − ≡ NVT Z Z − r = Δr, = 1, . , max Equivalently 1 1 N N ~ ~ = exp[ βU(r~ )]dr~ g(r) = 1 V δ(r12 r) N Q − N 〈 − 〉 〈 〉 NVT j<k Zrjk − X ∈I For a mixture: 1 N N = V exp[ βU(r~ )]dr~3 ... dr~N dr~12 QNVT 2 r − gj(r) = V δ(r~12 r~) Z 12 I Z 〈 − 〉 N ∈ Normalization (fluid): = ρ g(r)dr~ 2 ZI lim g(r) = 1 2 N ∞,r ∞ N → → g(r)ΔV NB: ideal gas at finite N: g(r) = 1 1/N (e.g., in periodic b.c.) ≈ 2V − The formula again: Number of particles around one chosen particle (in NVT): 2 g(r) = 〈N 〉 ρg(r~)dr~ = N 1 NρΔV ZV − 11/19 12/19 Correlation function and thermodynamics s08/3 Structure factor + s08/3 BB 1B For simple fluid with pair forces only: B B 1B B Residual internal energy: B k ν~ - inc 0 B 1 B B - B 1 βU 1 B U r e dr~ ... dr~ k ν~ B = j( j) 1 N inc − - B 〈 〉 QNVT B Z <j α X - - N V βU 2 U = e 4πr dr12(r12)dr~3 ... dr~N 2 Q − 12 〈 〉 NVT Z Distance of atom r~j from plane ν~ r~ = 0 is ν~ r~j (for ν~ = 1). · · | | N 2 Distance from wave source (wave plane of eq. ν~ r~ = const) via atom rj to the U = ρ (r)g(r)dr~ = 2Nπρ (r)g(r)r dr · 〈 〉 2 detector (ν~ r~ = const) is ν~ r~j ν~ r~j + const Z Z 0 · · − 0 · ~ Pressure: Wave vector kinc = kincν~, kinc = 2π/λ βP 2π 3 Formally the incident wave is (but phase) = 1 βρ g(r) (r)r dr ρ 3 0 − Z ~ exp[(kincν~ kincν~0) r~j] = exp[k r~j] − · · where ~ ~ k = kincν~ kincν~0, k = k = 2 sin(α/2)kinc αkinc − | | ≈ 13/19 14/19 Structure factor + s08/3 Structure factor and RDF + s08/3 3 Waves scattered by all atoms interfere: In a periodic box V = L ; k~ = 2πn/L~ : N N 1 N N ~ ~ ~ ~ ~ 2 K = exp[(kincν kincν0) rj] = exp[k rj], signal ∝ K S(k~) = exp( k~ r~j) exp(k~ r~) − · · 〈| | 〉 j=1 j=1 *N j 1 1 − · · + X X X= X= Definition of the structure factor (pure compound): 1 N 1 exp k~ r~j exp k~ r~j 2 = + ( ) + ( ) N N *j< − · · + 1 X S(k~) = exp( k~ r~j) N 1 ρ g r~ cos k~ r~ dr~ * j=1 − · + = + ( ) ( ) X V · Z It is a function of vector k~, we may imagine it in the periodic b.c. where k~ = 2πn/L~ , ~ ~ The integral diverges as V ∞ we subtract 0 = V cos(k r~)dr~, k = 2πn/L~ n~ Z3. → ⇒ 2 · (other method: convergence factor exp( αk ), limα 0) ∈ − R → Similarly, a 3D RDF is (not exactly isotropic even for a liquid) V ∞ S(k~) = 1 + ρ [g(r~) 1] cos(k~ r~)dr~ →= 1 + ρ [g(r~) 1] cos(k~ r~)dr~ N N 1 ( ) ZV − · Z − · g(r~12) = − ... exp[ βU(r~1, r~2,..., r~N)] dr~3 ... dr~N ρ2Q NVT Z Z − 15/19 16/19 Structure factor and RDF + s08/3 Structure factor: Isotropic system s08/3 Since g(r~) = g( r~), we have − ~ ~ ~ ~ ~ ~ S(k) 1 = ρ [g(r) 1] cos(k r)dr ~ ~ k r~ − − · S(k) 1 = ρ [g(r~) 1] cos(k r~)dr~ = ρ [g(r~) 1]e− · dr~ Z − − · − Z Z Let us assume k~ = (0, 0, k) and use spherical coordinates (ϕ, θ, r), then k~ r~ = kr cos θ and the 3D inverse Fourier transform is and · 1 ~ ~ k~ r~ ~ g(k) 1 = [S(k) 1]e dk ~ 8π3ρ · S(k) 1 = ρ [g(r~) 1] cos(k r~)dr~ (subst. z = cos θ) − Z − − Z − · Once again using S k~ S k~ : ∞ 2π 1 ( ) = ( ) 2 − = ρ r dr dϕ [g(r) 1] cos(krz)dz 1 0 0 1 − g k~ 1 S k~ 1 cos k~ r~ dk~ Z Z Z ( ) = 3 [ ( ) ] ( ) 4πρ ∞ − − 8π ρ − · Z = r[g(r) 1] sin(kr)dr k 0 − k = 0 limit: Z In the same way we can evaluate the inverse Fourier transform NVT : S(0)= 0 1 ∂ρ ~ ~ ~ ~ μVT : S(0)= 1 + ρ [g(r~) 1]dr~ = kBT (compressibility equation) g(k) 1 = [S(k) 1] cos(k r~)dk ∂p − 8π3ρ − · Z − T Z 1 ∞ = k[S(k) 1] sin(kr)dk 2π2rρ Z0 − 17/19 18/19 S(k) from simulations + s08/3 Structure factor + s08/3 S(k~) in simulations can be calculated directly by the same code as the k-space part 4 40 of the Ewald summation. If needed, we may sphericalize it: 3 30 S(k) = S(k~)/ 1 ~ ~ kX= k kX= k | | | | S(k) 2 S(k) 20 S(k) can be obtained by the inverse Fourier transform of g(r) If one wishes, 3D S k~ can be obtained by the inverse Fourier transform from RDF ( ) 1 10 g(r~) and vice versa Simulation-based S k for a mixture (where individual site-site RDFs need not be ( ) 0 0 available) allows for a direct comparison with the experiment 0 20 40 60 0 20 40 60 S(k) and S(k~) may detect unwanted crystallization of a supercooled system k/nm-1 k/nm-1 liquid - - - glass crystal - - - crystallite 19/19 Structure factor for multiatomic systems + s08/3 Structure factor for a “mixture” of atoms (bj = coherent scattering length) ~ 2 2 Q(k) j bj S(k) = 1 + N〈| | 〉 − 2P j bj P Q(k~) = bj exp[ 2πk~ r~j/L] j − · X NbNJbJ S S , = J J J = 2 J ( Nb) X X N = number of atoms of type ( Nb = j bj) P P P.