Solid State Communications 146 (2008) 468–471

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Solid State Communications

journal homepage: www.elsevier.com/locate/ssc

Atomic ordering recognized by convergence characteristics of the radialdistribution functionYong Yang ∗, Masae Takahashi, Yoshiyuki KawazoeInstitute for Materials Research, Tohoku University, Sendai, 980-8577, Japan

a r t i c l e i n f o

Article history:Received 21 January 2008Received in revised form25 March 2008Accepted 4 April 2008 by E.G. WangAvailable online 9 April 2008

PACS:68.18.Fg05.20.-y07.05.Tp

Keywords:C. Non-crystalline structureD. Atomic orderingD. Radial distribution functionD. Simulation

a b s t r a c t

Convergence characteristics of the radial distribution function (RDF) in a condensed phase have beenstudied theoretically. Using a canonical ensemble, the reversible work theorem for the RDF is deduced ina new way, which interprets the reversible work in a fresh aspect. The radius of convergence of the RDFis defined. For periodic structures such as crystals, convergence of the RDF is strongly correlated withatomic ordering therein. This correlation is demonstrated by comparing the structural properties of SiO2crystal, with the RDF of Al doped SiO2 structure and amorphous SiO2.

© 2008 Elsevier Ltd. All rights reserved.

Amorphous materials like liquids and glasses are ubiquitousand have an immense impact on our daily lives, ranging from thewater that we drink to telecommunications fibers and beautifulglass artworks. Their atoms are not arranged regularly as in acrystal. Such disordered structures give birth to unique propertiesin contrast to crystalline materials. Numerous experiments andmuch theoretical work have been devoted to study the staticand dynamic properties of liquids and glasses [1]. In particular,intermediate and extended atomic ordering in network-formingAX2 glasses has been a center of interests for a long time [2–8]. Forexample, recent neutron diffraction experiments on ZnCl2, GeSe2,and GeO2 glasses have identified two length scales associatedwith intermediate and extended range atomic ordering, whichare at distances greater than the nearest neighbor separations [6,7]. To understand such atomic ordering in AX2 glassy structures,most previous research work is devoted to looking for thestructural origin of diffraction patterns such as the well-knownfirst sharp diffraction peak (FSDP) [3–5]. In spite of these efforts,the microscopic mechanism regarding medium-range atomicordering in amorphous structures is still left in debate [2–8]. Verylittle is known about the role of energetics in the formation of

∗ Corresponding author. Tel.: +81 22 215 2050; fax: +81 22 215 2052.E-mail address: yyang@imr.edu (Y. Yang).

0038-1098/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2008.04.004

such intermediate and extended range atomic ordering. Physicalprinciples underlying this phenomenon invite further exploration.

In this work, we address this problem by studying therelationship between convergence characteristics of the radialdistribution function (RDF), and atomic ordering in condensedphase materials. Based on the definition of the RDF in statisticalmechanics, we have obtained the remarkable reversible worktheorem in a way different from previous literature [9]. Atomicordering in a condensed phase is related to the convergence lengthof the RDF, and the reversible work at a distance r is defined by theenergy deviation from potential energies of the weight averagedconfiguration of the system under consideration. The results areapplicable to liquids, glasses and crystalline structures. Theoreticalresults are demonstrated by our density functional theory (DFT)calculations on the SiO2 (α-cristobalite) crystal, with comparisonto Al doped SiO2, and previous studies on SiO2 liquids and glasses.

The RDF g(r) describes how the density of surrounding mattervaries as a function of the distance from a distinguishing point. Fora particle at some point O in the volume, if ρ = N/V is the averagedensity, then themeandensity at a distance of r away fromOdiffersfrom ρ by the factor g(r) [9]. Experimentally, the RDF is relatedto the measured structure factor by the Fourier Transform [9]. Insystems consisting of mono-species particle, g(r) can be calculatedby ensemble average over all particles [10]:

g(r) =V

N2

⟨∑i

∑j6=i

δ(⇀r −

⇀r ij)

⟩=

V

N

∑j6=i

δ(⇀r −

⇀r ij)

Y. Yang et al. / Solid State Communications 146 (2008) 468–471 469

Fig. 1. (Color online) Schematic diagram for the calculation of the radialdistribution function (RDF) in a periodic unit cell. Separations between A particles(large balls) and B particles (small balls), A and A particles, B and B particles aredenoted by rab , raa , rbb , respectively. The RDF gab(r) is calculated by counting thenumber of B particles inside a sphere shell (centered at A particles) with a radius ofr and thickness 1r. Here α, β, γ are the angles formed between the unit cell vectors⇀b and

⇀c ,

⇀a and

⇀c ,

⇀a and

⇀b , respectively.

=1ρ

∑j6=i

δ(⇀r −

⇀r ij) (1)

where N is the number of particles, V is the volume, ρ = N/V ,⇀r is a

position vector starting from the particle i,⇀r ij =

⇀r j −

⇀r i. The Dirac

δ-function can be reduced to [11]: δ(⇀r −

⇀r ij) = δ(r − rij)/(4πr2),

then g(r) =∑

j6=i δ(r − rij)/(4πr2ρ). The quantities r = |⇀r |, rij =

|⇀r ij| = |

⇀r j −

⇀r i|. A similar convention is used for denoting the

modulus of vector variables in the contexts hereafter.For a multi-species component system, the RDF for a B particle

around an A particle is gab(r) =V

NaNb〈∑

a

∑b δ(

⇀r −

⇀r ab)〉 =

VNb

∑b δ(

⇀r −

⇀r ab) =

1ρb

∑b δ(

⇀r −

⇀r ab), which can be reduced to

gab(r) =1ρb

∑b

δ(r−rab)4πr2

, with Na, Nb being the number of A and Bparticles, respectively.

Here A and B can be the same type or different type particles. Inpractice, gab(r) is calculated by replacing the δ-function with 1/1rat the points where r = rab:

gab(r) =1ρb

nb(r) ·δ(r − rab)

4πr2≈

1ρb

·nb(r)

4πr21r

∣∣∣∣r=rab

. (2)

The quantity nb(r) is the average number of B particles inside asphere shell that centers at an A particle with a radius of r andthickness 1r, and rab is the separation of particles A and B, asshown in Fig. 1. The quantity 1r is also the variation step of theradius r, or the sampling interval of the RDF. If the separation oftwo non-zero data points is smaller than 1r, they are consideredas continuous points in the RDF curve. The value of 1r is smallbut finite such as 0.01 Å, which is the experimental resolution forstructural measurement. For example, the down limit wavelengthof X-ray is λ ∼ 0.1 Å, the photon’s momentum p = h/λ, and1p ∼ h/λ, then 1r ≥ (h̄/2)/(h/λ) = λ/(4π) ≈ 0.008 Å ∼ 0.01 Å.In soft X-ray diffraction, the accuracy limit will be a larger number[12].Convergence of RDF at large scale—From Eq. (2), the discrete rabin crystals will lead to isolated sharp peaks of gab(r), which issimilar to the δ-function in the local area. For liquids or glasses,the continuous rab leads to broadened peaks, corresponding to localcoordination shells of the particles. In a large area, or r → ∞, the

behavior of gab(r) is not so obvious by judgment from Eq. (2) as thatin the local area. For condensed phases consisting of mono-speciesparticles, the joint distribution for finding a particle (any one) atthe position

⇀r 1, and any other particle (in the N particle system) at

⇀r 2 is [9]

ρ(2/N)(⇀r 1,

⇀r 2) = N(N − 1)

×

∫d

⇀r 3

∫d

⇀r 4

∫d

⇀r 5 · · ·

∫d

⇀r N exp[−βU(

⇀r 1,

⇀r 2,

⇀r 3, . . . ,

⇀r N)]∫

d⇀r 1

∫d

⇀r 2

∫d

⇀r 3 · · ·

∫d

⇀r N exp[−βU(

⇀r 1,

⇀r 2,

⇀r 3, . . . ,

⇀r N)]

(3)

where U(⇀r 1,

⇀r 2,

⇀r 3, . . . ,

⇀r N) is the potential energy of the

canonical ensemble with a constant volume (NVT), which is acontinuous function of particles’ coordinates, and stays unchangedwhen the same type of particles exchange their coordinates, e.g.,U(

⇀r 1,

⇀r 2, . . . ,

⇀r i, . . . ,

⇀r j, . . . ,

⇀r N) = U(

⇀r 1,

⇀r 2, . . . ,

⇀r j, . . . ,

⇀r i

, . . . ,⇀r N). The integral for spatial coordinates is

∫d

⇀r i ≡∫

dxdydz =∫r2 sin θdrdϕdθ (i = 1, 2, 3, . . . ,N). β = 1/(kBT) > 0,

kB is the Boltzmann constant, T is the temperature. The RDF isexpressed as follows [9]:

g(r) = g(|⇀r 1 −

⇀r 2|) = ρ(2/N)(

⇀r 1,

⇀r 2)/ρ

2 (4)

ρ = N/V is the average particle density in a volume V .For solids and liquids, the geometric size of the system should

be limited, i.e.,⇀r i (i = 1, 2, 3, . . . ,N) sits in a closed space

⇀R (R, θm,ϕm). If the spherical coordinate of

⇀r i is (r, θ,ϕ), then 0 ≤

r ≤ R, 0 ≤ θ ≤ θm, 0 ≤ ϕ ≤ ϕm. From the mean value theorem

for integration, there exists a series of configuration points⇀

ξ i(i =

1, 2, 3, . . . ,N) in⇀R , such that

ρ(2/N)(⇀r 1,

⇀r 2) = N(N − 1)

VN−2 exp[−βU(⇀r 1,

⇀r 2,

⇀

ξ 3, . . . ,⇀

ξ N)]

VN exp[−βU(⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, . . . ,⇀

ξ N)]

=N(N − 1)

V2 exp[−β1U] (5)

and

g(r) = g(|⇀r 1 −

⇀r 2|) = ρ(2/N)(

⇀r 1,

⇀r 2)/ρ

2

= ((N − 1)/N) × exp[−β1U] (6)

where 1U = U(⇀r 1,

⇀r 2,

⇀

ξ 3, . . . ,⇀

ξ N) − U(⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, . . . ,⇀

ξ N).For the gab(r) inmulti-species systems, the coordinate exchange

should be done in the same type of particles, the term N(N − 1)changes to NaNb in Eq. (5), ρ2 changes to ρaρb in Eq. (4) and Eq. (6),and the relation

gab(r) = exp[−β1U] (7)

holds when A and B represent different types of particles.

The term U(⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, . . . ,⇀

ξ N) is the potential energy of

the configuration (⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, . . . ,⇀

ξ N), which is the weightaverage of all possible configurations with the weight factorof exp[−βU(

⇀r 1,

⇀r 2,

⇀r 3, . . . ,

⇀r N)]. The most stable configuration

with the lowest potential energy U has the largest configuration

weight and characteristics in (⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, . . . ,⇀

ξ N). The term

U(⇀r 1,

⇀r 2,

⇀

ξ 3, . . . ,⇀

ξ N) describes the correlation of two particlesat

⇀r 1 and

⇀r 2 at the presence of the other (N-2) particles. The

term 1U = U(⇀r 1,

⇀r 2,

⇀

ξ 3, . . . ,⇀

ξ N) − U(⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, . . . ,⇀

ξ N)

describes the deviation of the configuration (⇀r 1,

⇀r 2,

⇀

ξ 3, . . . ,⇀

ξ N)

away from (⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, . . . ,⇀

ξ N). When |⇀r 1 −

⇀r 2| is equal to or

470 Y. Yang et al. / Solid State Communications 146 (2008) 468–471

Table 1Unit cell parameters of SiO2 crystal, and the Al–SiO2 structure

a (Å) b (Å) c (Å) α (◦) β (◦) γ (◦)

SiO2 (α-cristobalite)Calculation 5.053 5.053 7.037 90.00 90.00 90.00Experiment 4.971a 4.971a 6.922a 90.00 90.00 90.00Al–SiO2 9.952 8.982 7.532 99.66 87.62 90.28

Here a, b, c denote the lengths of lattice vectors of the unit cell, and α, β, γ are the angles formed between the vectors.a Ref. [17].

larger than certain value rc, interactions between the two particlesare negligible (|1U| ≤ δUI , δUI is the intrinsic fluctuation amplitude

of U(⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, . . . ,⇀

ξ N)), U(⇀r 1,

⇀r 2,

⇀

ξ 3, . . . ,⇀

ξ N) converges to aconstant value UC . This is true in condensed phase matter such assolids and liquids where Coulomb interactions are dominant. Thequantity rc can be regarded as the correlation length of any twoparticles under consideration. In the case |

⇀r 1 −

⇀r 2| ≥ rc, one can

find at least two points⇀r 1 and

⇀r 2 that sit at

⇀

ξ i and⇀

ξ j (|⇀

ξ i −⇀

ξ j| ≥

rc), which can be safely relabeled as⇀

ξ 1 and⇀

ξ 2, and the others

relabeled as⇀

ξ 3 to⇀

ξ N . Then one has U(⇀r 1,

⇀r 2,

⇀

ξ 3, . . . ,⇀

ξ N) =

U(⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, ..,⇀

ξ N) = UC , and 1U = 0, ρ(2/N)(⇀r 1,

⇀r 2) = N(N −

1)/V2for the same type particles or ρ(2/N)(⇀r 1,

⇀r 2) = NaNb/V2 for

different type particles when β > 0. Then ρ(2/N)(⇀r 1,

⇀r 2) = N(N −

1)/V2≈ (N/V)2 = ρ2, g(r) = ρ(2/N)(

⇀r 1,

⇀r 2)/ρ

2= 1 when N � 1;

or gab(r) = ρ(2/N)(⇀r 1,

⇀r 2)/(ρaρb) = 1 for different type of particle.

The results in Eqs. (6) and (7) are equivalent to the reversiblework theorem [9], which states that g(r) = e−βw(r), based on theprecondition that g(∞) = 1. The difference is that, for the sametype of particle, one has w(r) − 1U = kBT ln(N/(N − 1)), whichapproaches 0 and w(r) = 1U when N >> 1; for different typeparticles, w(r) ≡ 1U. Using Eq. (6) or Eq. (7), one can also estimatethe value of 1U from g(r).

Convergence of g(r) indicates the mean density of particlescontained in any sphere shell with a thickness of 1r and aradius r ≥ rc will equal to the average density ρ. Therefore,the quantity rc serves as the radius of convergence of the RDF.On the other hand, the value of rc reflects the atomic orderingin the system under consideration. The increase of disorder will

make it easier to find configurations (⇀r 1,

⇀r 2,

⇀

ξ 3, . . . ,⇀

ξ N) with

the same potential energy as U(⇀

ξ 1,⇀

ξ 2,⇀

ξ 3, . . . ,⇀

ξ N) = UC .This leads to the decrease of rc, beyond which the function

U(⇀r 1,

⇀r 2,

⇀

ξ 3, . . . ,⇀

ξ N) = UC , which belongs to a set of surfacesof constant energy. The difference between crystalline structuresand amorphous structures is the value rc of crystals is much largerthan the rc of amorphousmaterials, reflecting the long-range orderof crystalline structures. Because the entropy S, the RDF g(r) andpotential energies U can be expressed by the partition functionof the system [9], it is possible to build a quantitative connectionbetween entropy and the convergence radius of the RDF, althoughthis is outside the scope of presentwork. Even so, by comparing theconvergence characteristics of the RDF, it is feasible to recognizeatomic ordering in materials of the same chemical components.

To demonstrate the above viewpoint that atomic ordering canbe recognized by the convergence characteristics of the RDF, wego further to study the structural properties of α-cristobalite SiO2crystal by DFT calculations, and then compare the RDF gSiO withthe Al doped SiO2 structure (abbreviated as Al–SiO2), and previousresults on amorphous SiO2. Calculations were performed by theVASP code [13], employing a plane wave basis set and PAWpotentials [14] to describe the behavior of electrons. Exchange-correlation energies are described by Perdew-Wang (PW91)

functionals with generalized gradient approximations (GGA) [15].The unit cell of α-cristobalite crystal contains 12 atoms: 4 Si and 8O atoms. The unit cell of Al–SiO2 consists of 40 atoms: 12 Si, 4 Al,and 24 O atoms. Al–SiO2 is constructed by replacing some Si atomsin the host lattice of α-cristobalite with Al atoms and removing theexcess O to get the correct chemical composition for SiO2 and thenrelax the structure. The energy cut-off for plane waves is 520 eV.TheMonkhorst-Pack grid [16], with the origin being at the Γ point,has been used for integration in the Brillouin zone. A k-mesh of16 × 16 × 16 is used for the α-cristobalite crystal, and a k-meshof 6 × 6 × 6 is used for the Al–SiO2. After structural relaxation,the obtained cell parameters are listed in Table 1, the ones for theα-cristobalite crystal agree well with experimental data [17]. Infirst principles calculations, the total energy of a charge-neutralperiodic unit cell is evaluated accurately [18] without restrictionof the minimum image convention [10]. Thus the RDF of periodicstructures can be calculated in arbitrary large scale using periodicboundary conditions.

Fig. 2 shows the gSiO of the α-cristobalite and the Al–SiO2 inthe periodically extended structures of their unit cells. The uppanels, Fig. 2(a) and two insets, Fig. 2(b) and Fig. 2(c), describethe characteristics of gSiO in the α-cristobalite crystal. In Fig. 2(a),gSiO(r) shows a tendency of converging to 1 and the deviationfrom the value 1 is decreased as RSiO increases. Local featuresof gSiO are shown in Fig. 2(b), from 0 to 10 Å. The sharp peaklocated at 1.62 Å corresponds to the length of Si–O bonds in theα-cristobalite crystal. This agrees quite well with the experimentalvalue (∼1.61Å) [19]. FromRSiO = 280Å to 300Å, the gSiO fluctuatesaround 1 with an amplitude of ∼0.2, as shown in Fig. 2(c). Thusthe convergence length of gSiO of α-cristobalite crystal (≥300 Å)is much larger than that of amorphous and liquid SiO2, which is6–10 Å [20,21]. Down panels, from Fig. 2(d) to Fig. 2(f), displaythe features of the RDF gSiO in the Al–SiO2 structure. Comparedwithα-cristobalite crystal, the gSiO of the Al–SiO2 structure shows amuch smaller deviation from 1 and better convergence at the sameRSiO. Shown in Fig. 2(f), the gSiO of the Al–SiO2fluctuates around 1with a much smaller amplitude, which is ∼0.05. A smaller radiusof convergence of the RDF in the Al–SiO2is thus expected. Thisis due to lower atomic ordering of Si–O in the Al–SiO2 structure,which is reflected in the triclinic unit cell (Table 1), and thedistributed Si–O bond lengths (1.56–1.80 Å) shown in Fig. 2 (e).The same tendency and results are obtained for calculations withdifferent but reasonable1r, except that the height of the RDFpeaksmight be changed. With a higher temperature, less fluctuationand better convergence, and thus smaller radius of convergenceof the RDF will be expected, due to enhancement of thermalmotion of the atoms and dynamical broadening of peaks, whichwill lower or smear out local fluctuations between neighboringpeaks. Although the technique discussed above applies mainly intheory, it is possible to compare directly the convergence behaviorof the RDF at distances around 200–300 Å in experiments, by usingexperimental tools such as small-angle X-ray diffraction [22].

In conclusion, we have studied the convergence characteristicsof the radial distribution function (RDF) in condensed phase sub-stances. The statistical thermodynamic foundation underlying themedium-range atomic ordering in glassy structures is discussed.

Y. Yang et al. / Solid State Communications 146 (2008) 468–471 471

Fig. 2. (Color online) Convergence characteristics of the radial distributionfunctions (RDF) gSiO of the α-cristobalite crystal and the Al–SiO2 structure in aperiodically extended cell. The RDF are calculated using a sampling interval of1r = 0.01 Å. The up panels (a)–(c) are the RDF of the α-cristobalite, and thedown panels (d)–(e) are the RDF of the Al–SiO2 . All the vertical axes of the figuresrepresent gSiO , and horizontal axes represent the Si–O atomic separation.

Starting with a basic definition of the RDF in statistical mechan-ics, we have deduced the reversible work theorem for the RDF in anew way. The correlation length of two particles in solids and liq-uids is attributed to the deviation from the potential energy of theensemble averaged configuration, which defines the convergenceradius of the RDF, and interprets the reversible work in a new as-pect. Higher atomic ordering results in larger radius of convergenceof the RDF, which can be judged by large scale features of the RDF.

The relationship between convergence characteristics of the RDFand atomic ordering is further demonstrated by our density func-tional theory calculations on the SiO2 crystal, with comparison toAl–SiO2 structure and previous RDF results of amorphous SiO2. Inaddition to comparing entropy and counting the number of sym-metry operations, our studies provide a new approach for measur-ing atomic ordering in condensed phase materials, especially forstructures with medium/long-range order, and of the same chem-ical composition.

Acknowledgements

We are grateful to the staff of the Center for ComputationalMaterials Science at Institute of Materials Research of TohokuUniversity. One of us (Y. Y.) is supported by the Center for SpecialField Research, Tohoku University. We also thank Mr. Hiroshi Abe(Sekisui Chemical Co., Ltd) and Dr. Jiawei Wang (Argonne NationalLaboratory) for their helpful discussions.

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