January 8, 2004

Absence of decay in the amplitude of pair distribution functions at large distances

V. A. Levashov,∗ M. F. Thorpe, and S. J. L. BillingeDepartment of Physics and Astronomy, Michigan State University, East Lansing, MI 48824

The pair distribution function (PDF) that can be obtained by Fourier transformation of powderdiffraction data, traditionally was used to describe short range correlations in atomic positions.Amplitudes of peaks in experimental PDF decay as distance increases. Thus it was always assumedthat PDF at large distances is featureless. Puzzled by the observation that PDF calculated forcrystalline materials does not decay at large distances, if instrumental resolution is ignored, westudied the behavior of PDF at large distances. To the best of our knowledge, the origin of thePDF decay at large distances was never carefully discussed. It turns out, surprisingly, that increasein the number of neighbors at large distances does not lead to the decay of PDF independentlyfrom the type of the material. In other words PDF calculated with respect to one atom does notdecay at large distances not only for the crystalline, but also for amorphous materials. We foundthat this behavior in amorphous materials is caused by random fluctuations in the radial numberdensity. Thus PDF in amorphous materials decays mostly due to ensemble averaging over differentcentral atoms. We achieve accurate quantative description of fluctuations for the case when atomsdistributed randomly in space. Differences with amorphous case are discussed. The case of perfectsingle component crystals is the most interesting because in it all atomic positions are equivalentand there is no need to perform averaging over different atoms. Thus total measurable PDF forperfect crystals does not decay at large distances if instrumental resolution is ignored. Howeverthe origin of this behavior in crystals is significantly more complicated and this behavior of PDF isrelated to the still unsolved circle problem that C. Gauss formulated more than hundred years agoand that gave rise to the whole area in mathematics called lattice point theory. Further investigationof this case is obviously needed. For generality we discuss the case of PDF in d-dimensional space.Our results can be used to measure the amount of dislocations in crystalline materials and to testinstrumental resolution in scattering experiments.

PACS numbers: 100.100.AB

I. INTRODUCTION

PDF or radial distribution function G(r) has been usedto study atomic structures of materials since 19271,2. Itis a real function of a single real variable: radius r. Peaksin PDF occur at values of r =< rij > that correspondto the average distances between atomic pairs. Thus inprincipal PDF contains rather limited information aboutthe structure of the materials. Because of it PDF is usedmostly to characterize local atomic environment in ma-terials with the absence of long range order like amor-phous materials or liquids. Importance of PDF functionis caused by the fact that PDF is related in a simple wayto the scattering intensity in X-ray or neutron diffractionexperiments3.

If a material consists of atoms of only one type thenexperimental PDF Gex(r) can be obtained from reducedscattering intensity F (q) ≡ q[S(q)−1] via the sin Fouriertransformation:

Gex(r) =2π

∫ ∞

0

F (q) sin(qr) dq, (1)

where S(q) ≡ I(q)Nf2 , I(q) is scattering intensity averaged

over the directions of momentum transfer vector q, f isthe (q dependent) atomic form factor in x-ray scatter-ing, and is the (constant) scattering length in neutronscattering. The total number of scattering centers in the

sample N in practice is defined in such a way that S(q)goes to unity as q goes to infinity3.

As it was already mentioned PDF is related to struc-ture of a material. If material consist of only one type ofatoms then:

G(r) = 4πr[ρ(r)− ρ0], (2)

where the average number of atoms in a unit volume(average number density) is ρ0 and the number of parti-cles in the spherical annulus of thickness dr is given bydN(r) = 4πr2ρ(r)dr. It is clear that radial density ρ(r)should oscillate around ρo. In order compare modeledstructure with real structure of a material experimentmodeled ρ(r) usually is calculated as follows.

It is a widely used approximation justified by theDebye-Waller theorem, that in solids, if ~ri and ~rj areequilibrium positions of atoms i and j, the density withrespect to atoms i created by atom j is given by3:

ρij(~r) =1

(2πσ2ij)

32

exp[(~r − ~rij)2

2σ2ij

] (3)

where ~rij = ~rj − ~ri and σ2ij =< (~rij− < ~rij >)2 > is the

mean square deviation of ~rij from its equilibrium value< ~rij > due to atomic vibrations. In order to obtain ρ(r)it is necessary to perform angular averaging of ρij(~r).

2

0 10 20 30 40

Momentum Transfer q, (Å-1

)

0

100

200F

(q)

0 50 100 150

Distance r, (Å)

-20

-10

0

10

20

30

40

G(r

)Experimental F(q) from Ni at 15 K

Experimental G(r) from Ni at 15 K

FIG. 1: Experimental F (q) and G(r) of Ni at 15K measuredby neutron scattering.

This can easily be done in 3d leading to:

ρij(r) =exp [− (r−rij)

2

2σ2ij

]− exp [− (r+rij)2

2σ2ij

]

4πrrij

√2πσ2

ij

(4)

Since in solids deviations of atoms from their equilibriumpositions are much smaller than interatomic distancesσij rij , the second term in the numerator can be ig-nored and rij in the denominator can be substituted withr. The total radial density with respect to the atom i isobtained as a sum of contributions from different atomsj:

ρi(r) ∼=1

4πr2

∑j 6=i

exp[− (r−rij)2

2σ2ij

]√2πσ2

ij

(5)

Finally in order to obtain ρ(r) it is necessary to averageρi(r) over all sites N in the sample:

ρ(r) =1N

N∑i=1

ρi(r). (6)

PDF G(r) is obtained by substitution of ρ(r) into (2)3–7.From formulas (2,5,6) follows that peaks in G(r) occur atvalues of r that correspond to the average interatomic dis-tances in a material and that since ρ(r) oscillates aroundρo PDF should oscillate around zero.

By comparing PDF calculated for some model struc-ture with experimental PDF obtained from scatteringdata it is possible to verify how close the model structureis to the real structure of the material. PDFs for amor-phous materials and liquids contain only a few peaks,whose amplitudes quickly decrease as r increases, as it isshown in panel C of Fig.4. Because of this, little atten-tion was paid to the behavior of PDF at large distances.

In the last 20 years PDF has also been used to extractinformation about interatomic interactions in crystalline

materials8. This can be done because σij is interaction-dependent6,7. Experimental PDF for crystalline materi-als decays significantly slower than for amorphous ma-terials, as shown in Fig.1 that shows reduced scatter-ing intensity F (q) obtained in the neutron diffractionexperiment at the NPDF diffractometer at Los AlamosNational Laboratory and corresponding PDF, G(r), ob-tained from F (q) by Fourier transformation (1).

However traditionally there still was interest in be-havior of PDF at small distances because that is theregion from which structural information can be easilyextracted. Thus it seems to us, to the best of our knowl-edge, that although PDF was used to characterize ma-terials for almost a 100 years, behavior of PDF at largedistances was never carefully discussed. For example, itis unclear why PDF decays at large distances. In fact,there can be three different reasons that can lead to thedecay of PDF with an increase in distance.

The First and the most natural reason is the following.Note that from (5) follows that the value of ρi(r) at ris determined by those atoms that are in σ-vicinity ofthe spherical shell of the radius r. In other words, inorder for the site j to contribute to the value of ρi(r) thefollowing condition should hold: |r− rij | ∼ σ (see Fig.2).The volume that these atoms occupy can be estimated asdVσ,r ∼ 4πr2σ, while the number of atoms in this volumecan be estimated as dNσ,r ∼ 4πr2ρ(r)σ. As r increasesboth dNσ,r and dVσ,r increase and it is natural to expectthat ρ(r) ∼ dNσ,r/dVσ,r converges to ρo leading to thedecay of peak amplitude in PDF (see (2)). However thequestion of how ρ(r) converges to ρo was never properlyaddressed and we will see that prefactor r in (2) plays avery non-trivial role.

The Second reason that can lead to the decay of PDFwith increase of r is averaging (see (6)) of ρi(r) over dif-ferent central atoms i each of which can have a differentatomic environment.

Finally the third reason that can lead to the decay ofexperimental PDF is finite instrumental resolution thatincreasingly broadens the width of the peaks with an in-crease of the distance. In order to achieve better agree-ment with an experiment, PDF calculated according to(2,5,6) is usually convoluted with so-called instrumentalresolution function that is r dependent6. This convolu-tion increases the width of the peaks at large values of rmore significantly than the width of the peaks at smallvalues of r.

It is clear that finite instrumental resolution leads tothe decay of PDF, but do both the first and second rea-sons also lead to the decay of PDF? In this paper we aretrying to discuss this issue carefully. This study was ini-tiated by the observation (a puzzle) that PDF calculatedfor the fcc structure of crystalline Ni, in assumption of in-finite instrumental resolution (no convolution was made),does not decay at large distances at all. The average am-plitude of peaks in PDF function persists up to the verylarge distances as shown in Fig.3.

All atomic positions in the fcc structure of Ni are equiv-

3

FIG. 2: Only those sites contribute to the value of Gi(r) thatare in σ vicinity of the distance r with respect to site i.

0 5 10 15 20-2

0

2

100 105 110 115 120-2

0

2

1000 1005 1010 1015 1020-2

0

2

10000 10005 10010 10015 10020

Distance r, (Å)

-2

0

2

Pai

r D

istr

ibut

ion

Fun

ctio

n G

(r)

FIG. 3: PDF of Ni at 15K at different distances. The bluesolid lines are results of calculations while the red dotted lineson the two panels on top show the result of neutron scatteringexperiment taken from Fig. 1

-3

0

3

-3

0

3

Pai

r D

istr

ibut

ion

Fun

ctio

n G

(r)

0 10 20 30

Distance r, (Å)

-3

0

3

A

B

C

FIG. 4: PDF for the amorphous Si: ao ' 2.4 A, σ = 0.1A. A: with respect to 1 atom, B: averaged over 10 atoms, C:averaged over 20000 atoms.

alent. Because of it there is no need to perform averagingover different atomic sites (6). Thus it turns out, surpris-ingly, that increase in the volume of the spherical annuluswith r does not lead to the decay of PDF (first reasonmentioned above). That is something counterintuitive.However it basically means that the amplitude of fluc-tuations in ρ(r) − ρo calculated with respect to the oneparticular atom in fcc structure decays at 1/r (see (2)).The question is why it decays as as 1/r. The answeron this question for fcc lattice and other lattices is notcompletely known. This problem is related to the area ofmathematics called ”lattice point theory”9–11. The prob-lem that is basically equivalent to the one under consid-eration was originally formulated by C.F. Gauss12. Itwas studied for more than a hundred years and it is stillnot completely solved.

In our study we took a somewhat different approach.Since in nature, besides crystalline there are also amor-phous materials we decided to study at first structureswith disordered distribution of atoms in them. We afound simple exact solution for the case when atoms aredistributed completely randomly. Randomly means thatposition of two atoms can even coincide. The case ofamorphous materials is different from the random case,in particular, because there is excluded volume aroundevery atom. We found a way to take into account therole of the excluded volume. These two cases allow con-sideration in d-dimensional space and we follow this line.The case of crystals is the most complicated and we dis-cuss some aspects of PDF behavior on crystals at theend.

Thus experimental PDF obtained on a crystal decaysdue to finite instrumental resolution and imperfectionsof the crystalline structure. In amorphous materials the

4

decay of G(r) occurs primarily due to the averaging (6)of PDFs over many atoms that have different atomic en-vironment as shown in Fig.4.

For simplicity we discuss everywhere the case of mate-rials consisting atoms of one type. It is evident how tomake a generalization for the case of materials that areformed by different atoms.

Mean square deviations of the distance between a pairof atoms from its equilibrium value σ2

ij depends on posi-tions of the atoms in the sample. For a given structurethe values of σ2

ij can be calculated from a particular forcemodel4,5, as it was made for the top two panels of Fig.3.However if the distance between atoms i and j is largethen their motion is uncorrelated and it can be assumedthat σ2

ij is the same for all atoms that are far away fromeach other σ2

ij = σ2, as we do in this paper.The chapter is organized as follows. At first we define

PDF in d-dimensional space. Then we consider the caseof completely random atomic distribution. After thatwe discuss the case of the random distribution of atoms,with randomness limited by excluded volume. Behaviorof PDF in crystals is discussed after that. We conclude bydiscussing how reduced scattering intensity F (q) wouldlook like on a perfect crystal if there would be infiniteinstrumental resolution. In the main body of the paperthere are only simple evaluations with results of numeri-cal calculations (on square and triangular lattices for 2dand simple cubic, orthorhombic and fcc lattices in 3d),while more complex derivations are presented in the ap-pendices.

II. PDF IN D-DIMENSIONAL SPACE

A. Continuous Definition of PDF

It is possible to generalize the definition of PDF in 3dto the general case of d-dimensions. As it will be clearfrom the following PDF in d-dimensional space should bedefined as:

Gd(r) = rd−12 [ρd(r)− ρo]. (7)

This definition differs by the constant factor (4π) fromthe definition of G(r) in 3d given by (2). It is not impor-tant for us whether we define PDF with this prefactor orwithout. For the purpose of generality everywhere belowwe will use definition of PDF given by (7).

The angular averaging of (3) leading to (4) cannot beperformed in 2d or generally in d-dimensional space in aclosed form. However we will assume that forms similarto (5) with correct prefactors are valid for any dimension:

ρdi(r) =1

Ωdrd−1

∑j 6=i

exp[− (r−rij)2

2σ2ij

]√2πσ2

ij

, (8)

where Ωd is the total solid angle (Ω1 = 1, Ω2 = 2π,

Ω1 = 4π). In order to obtain ρd(r) from ρdi(r) averagingover different points (6) should be performed.

In the following we will also discuss PDF calculatedwith respect to a particular site:

Gdi(r) = rd−12 [ρdi(r)− ρo]. (9)

We will see in the following that the period and amplitudeof oscillations in Gdi(r) at large values of r are almostindependent from r for the given type of the structure,as it can be seen from Fig.3. The period of oscillations inGdi(r) is determined basically by σ. Thus one can expectthat average value of the integral:

< G2di(r) >=

1R2 −R1

∫ R2

R1

G2di(r)dr. (10)

calculated over a big enough interval (R1;R2) is inde-pendent of R1 if R1 is big enough. This approach will beused further for all types of materials.

For the case of amorphous materials instead of integra-tion of G2

di(r) over some range one may want to averageG2

di(r) over the different sites i. For the case when sitesare distributed in space completely randomly it is alsopossible to do averaging over different distributions. Allthese methods should, as it will be shown, lead to thesame result.

For a particular type of the lattice < G2di(r) > can

depend only on the lattice parameter a (i.e. densityρo = 1/λd) and the value of σ. Thus the dimensionlesscombination of < G2

di(r) >, ρo and σ which is:

g2di(σ/λ) =< G2

di(r) >σ

ρo, (11)

can depend only on dimensionless combination of ρo andσ i.e. σd/rhoo or on σ/λ. The same is true for the par-ticular type of the amorphous material or for the randomdistribution.

In the following instead of g2di(σ/λ) we will use the no-

tations g2d(L) for lattices, g2

di(R) for random distributionsof points, g2

di(F ) for random distributions of points withexcluded volume and g2

di(G) for distributions of pointsthat model glasses. Sometimes, when it is obvious whatis meant, we will drop out indices d and i or both withoutnotice.

B. Definition of PDF Through Bins

Definition of PDF given in the previous section is acontinuous definition of PDF. This definition is usefulbecause it allows us to compare modeled and experimen-tal PDFs. For our purposes, it is convenient to give adifferent definition of PDF also.

Let assume that we want to calculate PDF with respectto site i. Then we can define radial density with respectto site i at distance r by counting the number of points inthe interval (bin) [r − δ/2, r + δ/2). If there are Ni(δ, r)

5

0 5 10 15 20

-4

0

4

100 105 110 115 120

-4

0

4

1000 1005 1010 1015 1020

-4

0

4

10000 10005 10010 10015 10020

Distance r/a

-4

0

4

Pair

Dis

trib

utio

n Fu

nctio

n G

(r)

FIG. 5: PDF calculated for square lattice σ = 0.1a accordingto definitions through bins (blue lines) and Gaussians (redlines). Fig.1

points in the interval then radial density at r can bedefined, in assumption δ r, as:

ρdi(δ, r) =Ni(δ, r)Ωdrd−1δ

. (12)

It is clear that ρdi(δ, r) ≈ ρo. Then PDF is defined in thesame way as before (9) with ρdi(δ, r) instead of ρdi(r).

In order to make a comparison between continuous def-inition of PDF and definition of PDF through bins it isconvenient to establish some relationship between σ andδ. We find it convenient to define this relationship in as-sumption that in both approaches the weight of the peakassociated with every point is unity:

1 =∫ ∞

−∞

1√2πσ2

exp− (r − rij)2

2σ2dr = h · δ, (13)

where the height of the peak in definition through bins–h is forced to be the same as the height of the peakin continuous definition, i.e. h = 1/

√2πσ2. Thus δ =√

2πσ2.It was already shown (Fig.3) that amplitudes of peaks

in PDF calculated according to (8,9) for fcc (3d) struc-ture of Ni persist up to very high distances. Figure 5shows PDFs calculated according to both (continuousand through bins) definitions of PDF for square latticein 2d. Thus we see that both definitions are equivalentin some sense. We also see that the amplitude and fre-

quency of oscillations are basically the same at all dis-tances.

III. RANDOM CASE

It is rather difficult to understand behavior of PDFin cases when atoms or points form ordered structures–lattices. We found that the case when are points dis-tributed randomly, besides being easily understandable,also provides a great insight into the origin of non-decaying behavior of PDF.

The value of < G2di(r) > (see (10)) in the random case

can be easily estimated, if it is assumed that averagingis done over different distributions of sites j. If followsfrom the definition of PDF through bins (8,9) that:

< G2di(r) >=

< [Ωdrd−1δρdi(δ, r)− Ωdr

d−1δρo]2 >

Ω2dr

d−1δ2(14)

As follows from the definition of ρdi(δ, r) (12) the numberof particles inside the spherical annulus of radius r andthickness δ is given by Ni(δ, r) = Ωdr

d−1δρdi(δ, r), whilethe average number is given by Ni(δ, r) = Ωdr

d−1δρo.The well known result of statistical physics can beemployed14:

< [N −N ]2 >= N. (15)

Thus we get:

< G2di(r) >=

Ωdrd−1δρo

Ω2dr

d−1δ2=

1Ωd

ρo

δ(16)

Exact calculations for the continuous definition of PDFperformed (see text (10)) in appendices (A 1) in ensembleaveraging and in (A 2) in the integral approaches lead tothe following result:

g2di(R) =< G2

di(r) >σ

ρo=

12√

πΩd= const (17)

From (16,17) it follows that in the random case <G2

di(r) > does not depend on r. This result means thatan increase in the number of particles inside the sphericalannulus of thickness σ with r does not lead to the decayof PDF. It also means that in the random case ρi(r) con-verges to ρo as r

1−d2 , e.i. as 1/

√r in 2D and as 1/r in

3D.Note that for the random case the value of g2

di does notdepend on the value of σ/λ.

Fig.6 shows dependence of scaled PDF g3i(r) ≡G3i(r)

√σ/ρo on scaled distance r/σ. It shows that the

amplitude of oscillations in the scaled PDF g3i(r) doesnot depend on ρo or σ. It also shows that σ determinesthe period of oscillations. Note that the period T ∼ 4σis of the order of σ.

6

-0.3

0

0.3 ρ = 1.0 σ = 0.1

-0.3

0

0.3

Scal

ed P

DF,

r (ρ

i(r) -

ρo) (

σ/ρ o)1/

2

ρ = 0.1 σ = 0.2

0 25 50 75 100 125 150 175

Scaled Distance, r/σ

-0.3

0

0.3 ρ = 2.0 σ = 0.1

FIG. 6: Scaled PDF function g3i(r) ≡ G3i

√σ/ρo vs. reduced

distance r/σ.

IV. RANDOM CASE WITH EXCLUDEDVOLUME

A. General Discussion

There is an important difference between arrangementof atoms in real materials and arrangement of points inthe random case. In real materials two atoms cannotbe too close to each other. In other words, there is ex-cluded volume around every atom in which there can notbe another atom. Thus, in the random case, for a givenfinite density, the number particles inside the sphericalannulus can vary, in principle, between zero and infinity.In the amorphous case, in contrast, the maximum num-ber of particles inside the spherical annulus is limited byexcluded volume.

In real materials, the excluded volume around everyparticle and the atomic number density of the materialare closely related. However one can imagine a situa-tion when there is no direct relation between these twoquantities. For example, we can place particles in somebox randomly, but every new particle can be placed atsome point only if there is no any other particle withinsome distance. Suppose that we always want to placethe same number of particles in the box, so that densitywould be the same, but we want to vary the excludeddistance. If the excluded distance is small enough wewould be able to achieve our desirable density. However,as we would increase the excluded distance, it would bemore and more difficult to find the configuration of sitesthat would satisfy both requirements: given density andgiven excluded volume. But if the excluded distance istoo big we may not be able to achieve desired density.This problem is related to the so called problem of therandom close packing of spheres15–17.

FIG. 7: Distributions of points for two different excluded dis-tances. The left figure shows distribution that was obtainedby choosing excluded distance in such a way that excludedarea covers 1/4 of the total area. In the right figure, excludedarea covers 1/2 of the total area.

FIG. 8: Only those points that are in σ vicinity of the dis-tance r with respect to site i contribute to the value of Gi(r).Excluded volume around every point reduces the volume ac-cessible to the other points, that may want to enter the an-nulus due to fluctuations, and thus reduces the size size offluctuations in the number of points.

Figure 7 shows an example of two distributions of siteswith the same density in each of them. However, theexcluded distance in the left half of the figure was chosenin such a way that excluded area covers 1/4 of the totalarea. The right half of the figure shows distribution inwhich excluded area covers 1/2 of the total area.

As it was pointed out, the value of < G2di(r) > is deter-

7

mined by the average size of fluctuations in the number ofpoints inside the spherical annulus, i.e. < (N−N)2 >. Ifthere is excluded volume around every point, then for anypoint that wants to enter the annulus due to some fluctu-ation, not the whole volume of the annulus is accessible:it is reduced by excluded volume of those points that arealready in the annulus. Below we show that excludedvolume decreases the value of < g2

di(E) > compared withthe random case (17).

If the excluded volume around one site is V1f ∼ λd

and there are N ' ρoV sites inside the volume V , thenthe volume available for the placement of a new site isV − V1fN (in the random case it still would be V ).

Since, in order for the site j to contribute to the valueGdi(r), its distance from the site i should be ∼ r ± σ,we can say that site j should be within the volumeV ∼ Ωdr

d−1σ (see Fig.8). If inside this volume thereis already one site, then accessible volume is reduced byV1f ∼ γλd−1σ, where γ can be different for differentstructures. If there are N = ρoV sites inside the an-nulus then the accessible volume V ? is reduced by V1fNcompared with V , so that we have:

V ? = V − V1fN ∼ V − γλd−1σ · ρoV (18)

∼ V [1− γλd−1σ · 1λd

] = Ωdrd−1σ[1− γ

σ

λ]

In comparison with the random case for which we had(15): for the amorphous case we write:

< [N −N ]2 >= N?

= ρoV?,

so that (compare with (16)):

< G2di(r) >=

Ωdrd−1σρo[1− γ σ

λ ]Ω2

drd−1σ2

=1

Ωd

ρo

σ[1− γ

σ

λ].(19)

Thus the presence of excluded volume in the amorphouscase leads to the linear decay of < G2

di(r) > with anincrease of σ. A more profound derivation made in Ap-pendix (B) leads to (compare with (17)):

g2di(E) ∼= 1

2√

πΩd[1− γ σ

λ ]. (20)

Results (19,20) were obtained in assumptions that√(N −N)2 N? and that peaks corresponding to the

different sites do not overlap, i.e. σ/λ 1.

B. Results of Simulations in 2d

In order to verify numerically the role of excluded vol-ume in 2d, Np = 220 × 220 points were placed in thebox of size [−110, 110] in x and y directions. The trialcoordinates (xi, yi) of a particular point were generatedusing a random number generator. Point was placed at(xi, yi) if there was no any other point within the ex-cluded distance 2ξ. Otherwise a new attempt to gener-ate coordinates of the point was made. This procedure

0 0.1 0.2 0.3 0.4 0.5

Peak Width σ/λ

0

0.2

0.4

0.6

0.8

1

g 2i

2 (E)/

g 2i

2 (R)

0.05

0.10

0.20

0.525

FIG. 9: Dependence of g22i(E) and g2

2i(G) on σ for configura-tions with different excluded volumes. For the curves (withcircles) shown, excluded volume covers 0.05 , 0.10, 0.20 and0.525 of the total area of the box. The curve with squarescorresponds to the amorphous distribution of sites. It basi-cally coincides with the curve that has the fraction of excludedvolume 0.525. The curve with triangles corresponds to com-pletely random distribution.

was repeated until all Np points were placed in the box.In the beginning, when the box is almost empty, pointscan be placed basically anywhere. As the box fills up itbecomes more and more difficult to find suitable coordi-nates (empty space) in the box. The fraction of excludedvolume to the total volume of the box can be found, ifthe concentration c = 1/λ2, as Sf/Stot = πξ2/λ2. In oursimulations λ = const = 1.

Then, for the particular choice of ξ (distribution), forall sites that were inside the central box of size [−50, 50]in x and y directions the PDF G2i(r) was calculated fordifferent values of peak width σ. The average value ofg22i =< G2

2i(r) > σ/ρo was found by averaging over thepoints that lay inside the central box. Our simulationsshow that g2i2(E) does not depend on r, which is inagreement with (20). At larger values of r > 20 somesize effects can be observed similar to those that occur in3d (see description in 3d section).

Figure (9) shows the dependencies of the ratio of g22i(E)

to the constant exact (17) value g22i(R) for the random

case on σ/λ for values of excluded volume πξ2/λ2 equalto 0.05, 0.10, 0.20, 0.525.

It is interesting to compare results obtained from therandom case with excluded volume with results obtainedon some modeled amorphous structure. The curve withsquares represents the results that were obtained on 3-fold coordinated network constructed from the honey-comb lattice by amorphizing it with an amorphizationprocedure similar to WWW method18,19. The curvecorresponding to the amorphous case basically coincideswith the curve obtained on the random structure with

8

0 0.1 0.2 0.3 0.4 0.5

Fraction of Excluded Volume πξ2/λ2

0

0.05

0.1

0.15

0.2

Slop

e, γ

0 0.1 0.2 0.3 0.4 0.50

0.25

0.5

0.75

1

g 2i

2 (E)/

g 2i

2 (R)

2d

FIG. 10: Dependence of the slope (linear part) of the curvesg22i versus σ (see Fig.9) on excluded volume. The value of

excluded volume 0.525 is very close to the limiting accessiblevalue. The inset shows how the ratio of the asymptotic value(large σ/λ) of g2

2i to the value of g22i for the random case

depends on excluded volume. Note the scale on the y-axis.

the fraction of excluded volume 0.525. In order to obtainthe data for amorphous case to the central box of size [-25.0;25.0] in x and y directions that contained 800 pointsperiodic boundary conditions were applied. The averag-ing of G2

2i(r) in the interval of r ∈ (5.0; 15.0) was madeover the 800 points that are inside of the central box. Al-though this approach has obvious shortcomings (see theirdiscussion in 3d section), it seems, that obtained resultsare insensitive to them.

The curve with triangles shows the data with respectto the point at the origin while the others 10002 pointswere randomly distributed (zero excluded volume) in thebox of size [−1000.0; 1000.0] in x and y directions. ThePDFs were calculated in the interval of r ∈ (3.0; 7.0).The averaging of G2

2i(r) was made over 150,000 differentconfigurations/distributions in order to calculate g2

2i(R).As excluded volume increases, the absolute value of the

curve’s slop γ at small values of σ/λ also increases, whilethe asymptotic value of g2

2i(E) reachable at large values ofσ/λ decreases. We can see that g2

2i(E) decreases linearlywith σ/λ at small values of σ/λ, in agreement with (20).Figure 10 shows dependence of the slop γ and asymptotic(large σ) values of g2

2i(E) on forbidden volume.The precision with which we determine the slope is

about 5% of the value of the slope itself. Precisionwith which asymptotic value is determined is significantlylower ∼ 20%, since for the large values of σ/λ there occuronly few oscillations in G2i(r) on the interval of the stud-ied distances. We used as an asymptotic value of g2

2i(E)its value at σ/λ = 0.75.

The fact that g22i(E) does not decay to zero at large

values of σ/λ indicates that even in the case when thereis excluded volume, the number of particles inside thespherical annulus can fluctuate, although fluctuationsare limited. Due to the limited size of fluctuations, the

0 0.1 0.2 0.3 0.4 0.5

Peak Width σ/λ

0

0.2

0.4

0.6

0.8

1

g 3i

2 (E)/

g 3i

2 (R) 0.05

0.10

0.20

0.30

0.02

Glass

FIG. 11: Dependence of g23i(E) and g2

3i(G) on σ for configu-rations with different excluded volumes. For the curves (withcircles) shown, excluded volume covers 0.02, 0.05 , 0.10, 0.20and 0.30 of the total volume of the box. The curve withsquares corresponds to the distribution of sites in amorphousSi. It basically coincides with the curve that has the frac-tion of excluded volume 0.35 (not shown). The curve withtriangles corresponds to completely random distribution.

value of g22i(E) in cases when there is excluded volume is

smaller than for the completely random case. The biggerexcluded volume leads to smaller fluctuations and thus itresults in a smaller value of g2

2i(E). Behavior of g22i(E)

at large values of σ/λ corresponds to the situation whenpeaks that originate from different points overlap. Thatis the limit when amplitude of atomic vibrations becomescomparable with interatomic distances.

C. Results of Simulations in 3d

Numerical simulations similar to those made in 2d canalso be made in 3d. It is interesting to compare the re-sults obtained for the random distribution of points lim-ited by excluded volume with results obtained on somestructure that represents some real amorphous material.

As a structure that represents a real material, we usedthe modeled structure of amorphous Si20,21. In amor-phous Si the average distance between the nearest atomsis ∼ 2.5 A. Most of the simulations were made on thesample containing 20,000 atoms that occupy the cubicbox of size [−36.23; 36.23] Ain x, y and z directions. Pe-riodic boundary conditions were applied to this box. Thesquare of PDF G3i(r) was calculated for all atoms in theoriginal box. The average value g2

3i(E) was found by av-eraging of G2

3i(r) over 20,000 atoms. In order to find thedependence of g2

3i(E) on σ calculations were made for dif-ferent values of σ. The described method of calculationis not quite correct, because there are some predefinedcorrelations in atomic positions due to periodic bound-

9

0 0.1 0.2 0.3

Fraction of Excluded Volume (4/3)πξ3/λ3

0

1

2

3

4

5

Slop

e, γ

0 0.1 0.2 0.30

0.25

0.5

0.75

1

g 3i

2 (E)/

g 3i

2 (R)

3d

FIG. 12: Dependence of the slope (linear part) of the curvesg22i versus σ (see Fig.11) on excluded volume. The value of

excluded volume 0.525 is very close to the limiting accessiblevalue. The inset shows how the ratio of the asymptotic value(large σ/λ) of g2

2i to the value of g22i for the random case

depends on excluded volume.

ary conditions and the fact that contributions of somepairs of atoms to g2

2i(E) are counted more than once.However, we repeated some of our calculations on a big-ger sample containing 100,000 atoms without periodicboundary conditions. From that we conclude that theobtained results are not sensitive to the shortcomings ofthe used calculation procedure.

Coordinates of the points for the random case withexcluded volume were generated using a random numbergenerator by distributing 903 = 729, 000 points in the boxof size [-45.0;45.0] in x, y and z directions. A point wasplaced at generated coordinates if there was not any otherpoint within excluded distance. Otherwise another set oftrial coordinates was generated. Then from those pointsthat were in the smaller box of size [−20.0; 20.0] (ap-proximately 403 = 64, 000) 10,000 points were randomlychosen. With respect to them G3i(r) were calculated.We found that, as in 2d, the amplitude of oscillations inG3i(r) is distance independent, if size effects are ignored.PDF G3i(r) was averaged over these 10,000 points in or-der to find g2

3i(E). The value of g23i(E) is constant at

values of r < 10.0. At larger r values g23i(E) slowly in-

creases, which is the size effect caused by the finite size ofthe original box (finite number of points also limits thesize of fluctuations). The last statement was verified bycalculating g2

3i(E) from the bigger box [-90.0;90.0] in x,y and z with 1803 = 5, 832, 000 points in it with the sizeof the small box still [-20.0;20.0]in x, y and z directions.For this bigger box the increase in the value of g2

3i(E)was smaller.

Figure 11 shows the dependencies of the ratiog23i(E)/g2

3i(R) (where, in the ratio, the numerator rep-resents the value that was obtained from the randomdistribution with excluded volume and the denominatorrepresents the value that was obtained for the purely ran-

dom case) on σ/λ for different values of excluded volume.The curve that corresponds to the amorphous Si is alsoshown.

Figure 12 shows how the slopes at small values of σ/λand asymptotic values at large values of σ/λ of the curvesg23i(E)/g2

3i(R) depend on excluded volume (4πξ3)/(3λ3).

V. CRYSTALS

A. General Discussion

Behavior of the PDF at large distances in crystals isthe most intriguing. This case is also the most impor-tant because only in crystals can non-decaying behaviorof the PDF be observed experimentally. Thus, if a crys-tal is formed by atoms of one type and all lattice sitesare equivalent (in assumption that crystal structure isperfect) then there is no need to perform averaging (6)over different lattice sites since Gi(r) is the same for allof them, i.e Gi(r) = G(r). The case of crystals also turnsout to be the most complicated.

In this case we cannot say that points inside the annu-lus are randomly distributed, since points are fixed on thegrid. Thus we can not use the same philosophy that wasused for the random distribution of points to describethe case of crystals. However, as it will be shown, thegeometry of the circle leads to some similarities betweenthese two cases.

Behavior of PDF in crystals at large distance is relatedto complex and extensively studied mathematical area –so called lattice points theory. The problem is the moststudied in 2d. More than a hundred years ago C.F. Gaussformulated the following problem12,13: how many latticepoints of the square lattice are there inside the of circleof radius r with the center at one particular point?

On Fig.13 lattice points that lie inside the circle ofradius r are shown as small filled circles, while sites thatlie outside are shown as small open circles.

Assume that every site of the lattice is in the centerof the square with side a, as shown on the Fig.13. Thus,there is a correspondence between the area that lies insideof the circle of radius r and the number of lattice pointsinside of the circle. It is clear that the number of pointsinside of the circle can be written in the form N(r) =N(r) + h(r), where N(r) = π(r2/a2), and h(r) is theerror term.

It is easy to see that Gauss’s circle problem is relatedto the behavior of PDF. Let us use definition of PDFthrough bins (12). Then from the definition of PDF in2d (7), if δ r, it follows that:

G(r) = [2πrδρ(r)− 2πrδρo]/(2π√

rδ), (21)

while from definitions (12) of ρ(r) and ρo it follows that:

2πrδρ(r) = [N(r + δ/2)−N(r − δ/2)] (22)

and

2πrδρo = [N(r + δ/2)−N(r − δ/2)]. (23)

10

FIG. 13: Illustration for the Gauss’s circle problem. Sitesthat lie inside of the circle of radius r shown as black filledcircles, while sites that lie outside of the circle of radius r asopen circles. Only those sites contribute to the value of Gi(r)that are in σ vicinity of the distance r with respect to site i.

Thus:

G(r) = [h(r + δ/2)− h(r − δ/2)]/(2π√

rδ). (24)

The simplest estimation of h(r) made by C.F.Gauss12,13 follows from Fig.13. All squares that corre-spond to the sites that are inside of the circle of radiusr lie completely inside of the circle with radius r + d/2,where d =

√2a is the diagonal of the little square. From

the other side, the circle of radius r − d/2 is completelycovered by the squares that correspond to the sites thatlie inside of the circle of radius r. Thus, we concludethat:

1a2

π(r − a√2)2 ≤ π(

r

a)2 + h(r) ≤ 1

a2π(r +

a√2)2,

i.e:

−12(2πr

a) +

π

2≤ h(r) ≤ +

12(2πr

a) +

π

2. (25)

Formula (25) suggests that the value of the error termh(r) is determined by the length of the circumference:2πr. From (24,25) it follows that:

−√

2√

r

aδ≤ G(r) ≤

√2√

r

aδ. (26)

We see that the obtained borders for h(r) are veryweak. If h(r) would depend on r as h(r) ∼ r then G(r)would increase as

√r while numerical simulations sug-

gest, as follows from Fig.3 and as we will see further,that G(r) ∼ const meaning that in fact h(r) ∼

√r.

A number of works22–24 were devoted to the more pre-cise estimation of h(r). In 1990 M.N. Huxley showedthat22:

|h(r)| ≤ Crθ, (27)

where C is a constant and the bounds on θ are

12

< θ ≤ 4673≈ 0.63 (28)

The lower limit was obtained independently by G.H.Hardy and E. Landau in 191523,24.

If θ would be 1/2 it would be natural to expectthat amplitudes of oscillations in G(r), on average, aredistance-independent. Since θ > 1/2 it indicates that ex-treme amplitudes of peaks in G(r) grow with the increaseof distance r. One may argue that (27,28) basically putlimitations on the extreme values of h(r).

Thus (27,28) suggest that h(r) can exhibit, in principle,rather complicated distance-dependent behavior. How-ever, results of our numerical simulations suggest that,on average, amplitudes of peaks in G(r) are almost dis-tance independent. It means that θ is almost 1/2. Beingmore precise, it seems that, θ being just a little bit biggerthan 1/2 converges to 1/2 from above, as r →∞.

This observation is in agreement with result that wasobtained in 1992 by P. Bleher who showed (we are notgiving here precise formulation) that there exists a limit:

limR→∞

1R

∫ R

0

h2(r)r

p(r/R)dr = α2, (29)

where p(x) is an arbitrary probability density on x ∈ [0, 1]and α2 is a positive constant. This result suggests thatthere is also another limit that is important for us (see(24)):

1R2 −R1

∫ R2

R1

G2(r)dr = α2. (30)

In the following sections, after the discussion of the ex-actly solvable 1d crystal, we present some puzzling re-sults of our numerical simulations in 2d and 3d. Thenwe return to the general discussion again.

B. Exact Solution for 1d Crystal

According to definition of PDF in d-dimensions (7,8,9),in 1d:

G(r) = ρ(r)− ρo, (31)

where

ρ(r) =1√

2πσ2

∞∑n=−∞

exp [− (r − na)2

2σ2] (32)

and ρo = 1/a.

11

0 0.1 0.2 0.3 0.4

Peak Width σ/a

0

0.2

0.4

0.6

0.8

1

g 12 (L)/

g 12 (R)

FIG. 14: Figure shows the dependence of the ratiog21L(σ)/g2

1R(σ) on σ. At small values of σ, when differentpeaks in G(r) do not overlap, this ratio linearly decreases asσ increases. As peaks start to overlap (large values of σ), therate of decay in the average amplitude of PDF decreases.

It can be shown (see appendix (C)) that for any valueof σ:

g21L(σ)

g21R(σ)

= [1− 2√

πσ

a] + (33)

∞∑n=1

exp [−n2a2

4σ2]

This curve is plotted on Fig.14.The last term in (33) can be ignored at small values of

σ. As σ increases it changes the the rate of decay in theaverage amplitude of PDF. Thus, behavior of PDF for1d crystal is similar to the behavior of PDF in glasses.

C. Results of Simulations in 2d

In order to demonstrate that on average the ampli-tude of oscillations in h(r) indeed scales as r1/2 we cal-culated explicitly for the square lattice the number oflattice points inside the circle of radius r ∈ (0; 100, 000)awith step 0.01a, where a is the lattice spacing. We definefunction F (r) as25:

F (r) =N(r)−N(r)√

r. (34)

If h(r) ∼ r1/2 then the amplitude of oscillations in F (r)on average should be constant.

Insets on the Fig.15 show F (r) vs. r dependence inthree different intervals when r ∈ (0; 100, 000).

The values of F 2(r) were averaged over 100,000 differ-ent values in a few intervals of length 1000a. The averagevalue of < F 2(r) > as a function of r is shown on Fig.15as a dashed curve. Circles are plotted in the beginningand in the end of the corresponding averaging interval.From the vertical scale it follows that while r changesfrom 0 to 100, 000 the average value of F 2(r) changesby less than 1%. From this we conclude that basically

0 20000 40000 60000 80000 1e+05

Distance r/a

2.555

2.56

2.565

2.57

2.575

<F2 (r

)>

0 200 400 600 800 1000

-5

0

5

9000 9200 9400 9600 9800 10000

-5

0

5

F(r)

99000 99200 99400 99600 99800 1e+05

-5

0

5

FIG. 15: Inset shows how F (r) depends on r for r ∈(0; 1000)a, r ∈ (9000; 10, 000)a and r ∈ (99, 000; 100, 000)a. Itshows that on average amplitude of oscillations in F (r) doesnot depend on r. At least, it is impossible to see it by eye.The dashed curve on the main figure shows how < F 2(r) >averaged over few intervals of r of length 1000a depends onthe position of the averaging interval. At the beginning andat the end of every averaging interval circles are plotted.

h(r) ∼ r1/2. There is also a slight increase in the valueof < F 2(r) > with r. This increase becomes slower as rincreases. From that we conclude that θ converges fromabove to 1/2 as r increases.

An example of PDF calculated for the square lattice ina continuous approach is shown on figure (5) as a dashedline. PDF calculated for triangular lattice exhibits simi-lar behavior. In order to calculate g2

2(L) (see (10,11) forthe square and triangular lattices as a function of σ, thesquares of corresponding PDFs were integrated (see (10))in the interval of r ∈ (1000λ; 2000λ). We should pointout here again that the values of g2

2(L) are very insensi-tive to the position of the averaging interval. Calculationin the interval r ∈ (300λ, 600λ) leads to almost the sameresults.

Figure 16 shows how, for the particular type of thelattice, the ratio of g2

2(L) to the g22i(R), where g2

2i(R) isthe exact constant value in completely random case (see(17)) depends on σ/λ.

As it was already pointed out g22(L) being dimension-

less can depend, for the particular type of the lattice,only on a dimensionless combination of ρo and σ i.e. onρoσ

2 or on σ/λ (see (11) with related text). Thus thecurves presented on Fig.16 are universal for triangular

12

0

5

10

0 0.05 0.1 0.15 0.2 0.25 0.3

σ/λ

0

5

10

Square Lattice

Triangular Latticeg 22 (L

)/g 2i

2 (R)

FIG. 16: Dependencies of g22(L)/g2

2i(R) on σ/λ for triangu-lar and square lattices. Triangles and squares represent theresults of numerical calculations for triangular and square lat-tice respectively. The solid fitting curves are given by formulasg22(tri)/g2

2i(R) = 2.483(σ/λ)−1/3−18.264(σ/λ) for triangular

lattice and g22(sqr)/g2

2i(R) = 1.849(σ/λ)−1/314.478(σ/λ) forsquare lattice. Dashed lines highlight the effect of excludedvolume. Expressions for them are given by the first diver-gent term in the formulas for the solid curves. They do notcontain the second term (that might be) caused by excludedvolume. From the figure, it follows that the effect of the ex-cluded volume is bigger for triangular lattice than for squarelattice.

and square lattices.From (17,20,33), Fig.9,14 and Fig.16 it follows that

behavior of g22(L)/g2

2i(R) vs. σ in case of 2d crystals isvery different from its behavior in the completely ran-dom case or in the random case with excluded volume(the amorphous case too), or in the case of 1d crys-tal. In the random case this ratio is unity by definition(g2

2i(R) = const). In the random case with excluded vol-ume it decays from unity (at small values of σ) to somefinite smaller value as σ increases, as it does for 1d crys-tal also. However, in the case of 1d crystal this ratiodecays to zero. In 2d crystals g2

2(L)/g22i(R) diverges at

small values of σ and it seems that it is decaying to zeroat large values of σ.

We tried to fit numerical data with analytical curvesin the form:

g22d(L)

g22i(R)

= c1(σ

λ)−η − c2(

σ

λ), (35)

where c1, c2 and η are positive constants. The first termwas chosen in the simple form that can provide diver-gence of g2

2(L)/g22i(R) at small values of σ. The second

term originates from the notion that in the crystallinecases, in some sense, there is also excluded volume aroundevery lattice point. Thus it was chosen in the same formas it is in the random case with excluded volume or inthe amorphous case (20) . In fact the form (35) provides

a rather poor fit in 2d. However, the same form providesan extremely good fit in 3d case, as we will see.

For triangular lattice the values of coefficients that canprovide the best fit in the whole studied range of σ are:c1 = 2.483, η = 1/3 and c2 = 18.264. However, at smallvalues of sigma, different values of coefficients provide abetter fit: c1 = 3.742, η = 0.245 and c2 = 43.433.

The situation with the square lattice is similar. Thevalues of coefficients that provide the best fit in the wholerange of σ are: c1 = 1.849, η = 1/3 and c2 = 14.478. Thevalues of coefficients that provide a better fit at smallvalues of σ are: c1 = 2.806, η = 0.245 and c2 = 33.410.

Note that the value of the coefficients c2 for triangularlattice is bigger than for the square lattice. That is inagreement with the observation that triangular lattice ismore densely packed than square lattice. In other words,the ratio of excluded volume to the total volume of thesample for triangular lattice is bigger than for squarelattice. Thus from Fig.9 it follows that c2 for triangu-lar lattice should be bigger than for square lattice, as itis. This observation can be considered to be an indirectindication that the form of the second term in (35) iscorrect.

D. Results of Simulations in 3d

An example of PDF calculated for fcc lattice is shownon Fig.3. In order to obtain the values of g2

3(L) as afunction of σ the square of PDF was integrated for fccand orthorhombic (b/a = 2.0, c/a = 3.0) lattices in theinterval r ∈ (1000λ; 2000λ), where ρo = 1/λ3 (obtainedresults are insensitive to the position of the averaginginterval as in 2d).

Figure 17 shows how the ratio g23(L)/g2

3i(R) dependson σ for fcc and orthorhombic lattices. This ratio in 3dalso diverges as σ → 0, as it does in 2d. We tried to fitthe data obtained from numerical calculations with thefitting curves in the form similar to the one used in 2d(35). We found that curves:

g23(fcc)g23i(R)

= 0.855(σ

λ)−1 − 15.079(

σ

λ) (36)

for fcc lattice and

g23(ort)g23i(R)

= 0.326(σ

λ)−1 − 2.563(

σ

λ) (37)

for orthorhombic lattice provide an extremely good fit tothe numerically calculated data.

Note that the value of the coefficient c2 for fcc lat-tice is bigger than for rectangular lattice. It is, as in2d, in agreement with conception of excluded volume.Since fcc lattice is packed more densely than orthorhom-bic lattice, excluded volume for fcc lattice is bigger thanfor orthorhombic lattice. Thus according to Fig.11,12 itshould be c2(fcc) > c2(ort).

13

0 0.05 0.1 0.15

Peak Width σ/λ

0

10

20

30

40

50

60

70

80

90

g 32 (L)/

g 3i

2 (R)

5

10

15

0.05 0.1 0.15

2

4

6

fcc

orthorhombic

fcc

orthorhombic

FIG. 17: Results of numerical simulations for FCC and rect-angular lattices. For rectangular lattice b/a = 2 and c/a = 3.Points represent results of simulations. Solid fitting curvesare given by formulas described in the text. Dashed lineshighlight the effect of excluded volume. Expressions for themare given by the first divergent term in the formulas for thesolid curves. Thus they do not contain the second term (thatmight be) caused by excluded volume. It can be seen fromthe figure that the effect of excluded volume is bigger for FCClattice than for rectangular lattice.

We would like to highlight again that the curves plot-ted on Fig.17 are universal for a given lattice, i.e. theratio g2

d(L)/g2di(R) indeed can depend only on the ratio

σ/λ.

E. Speculations on the Origin of σ-Divergence.

In order to find g2d(L) it is necessary to calculate an

integral (10):

< G2d(r) >= 1

R2−R1

∫ R2

R1G2

d(r)dr (38)

= 1R2−R1

∫ R2

R1

[Ωdrd−1σρ(r)−Ωdrd−1σρo]2drΩ2

drd−1σ2

Since the value of the expression above (almost) doesnot depend on position of the averaging interval, thevalue of the last integrand should also be on average r-independent. We know that, in the case of random distri-bution of points, the fluctuations in the number of pointsinside of annulus is determined by the volume of the an-nulus or by the number of points in it (see (14,15,16)):

[Ωdrd−1σρ(r)− Ωdr

d−1σρo]2 ∼ Ωdrd−1σρo. (39)

Thus for the random case we would get:

g2d(R) =< G2

d(r) >σ

ρo∼ 1

Ωd= const (40)

In crystalline case, when lattice points are fixed on thegrid, it is natural to assume that the size of fluctuationsis determined by position of the surfaces enclosing theannulus and thus basically by the surface area of the an-nulus. In principle, fluctuation on internal and externalsurfaces can be correlated. The size of correlations canbe σ-dependent. Thus, for the case of crystals, we write:

[Ωdrd−1σρ(r)− Ωdr

d−1σρo]2 ∼ Ωdrd−1λρoη(σ/λ). (41)

So that for crystals instead of (40) we would get:

g2d(L) =< G2

d(r) >σ

ρo∼ 1

Ωd(σ

λ)−1ηd(σ/λ). (42)

Thus, if for the random case we had g2d(R) ∼ const, for

the ordered distribution of points, we can get divergenceif, for example, η(σ/λ) ∼ const.

Fitting curves for fcc and orthorhombic lattices in 3d(see Fig.36 with formulas (36,37)) suggest that η3(σ/λ) ∼const. In 2d in order to obtain divergence (σ/λ)−1/3 (seeFig.16) we should assume that η2(σ/λ) ∼ (σ/λ)2/3.

However, everything that is written in this sectionshould be considered a hypothesis (we are not specialistsin lattice point theory) and further investigation of thisrather complicated problem, in connection with Gauss’scircle problem, is obviously necessary.

F. Behavior of PDF at Large Values of σ

Another puzzle represents behavior of PDF in crystalsat values of σ comparable with interatomic spacing σ ∼a.

As an example, Fig.18 shows that somethimes thereoccur big oscillations in PDF at large values of σ forsimple cubic lattice. For other lattices in 2d and 3d thisbehavior is less pronounced.

G. Order and Disorder. Similarities andDifferences.

In behaviors of PDF on crystals, on completely randomdistribution of points, on random distribution of pointswith excluded volume and on glasses there are some sim-ilarities, as well as differences.

Thus, in all of these structures amplitudes of oscilla-tions in PDF that were calculated with respect to oneparticular site persist (or almost persist for the case ofcrystals) with an increase of distance. We understand theorigin of this behavior in the case of completely randomdistribution of points and in case of random distributionwith excluded volume. The case of glasses is very similar

14

-0.4

0

0.4

0 100 200 300 400 500

Distance (r/a)

-0.2

0

0.2

Pair

Dis

tibut

ion

Func

tion

G(r

)σ = 0.3

σ = 0.7

FIG. 18: Results of numerical calculations of PDF for simplecubic lattice at large values of σ. PDF on the top panel wascalculated with σ = 0.3, while on the bottom panel withσ = 0.7. One can see that there occur large oscillations inPDF at values of r/a = 50, 100, 150, etc.

to the case of random distribution with excluded volume.We do not really understand the origin of this behaviorin the case of crystals.

We also understand the dependence of the ratiog2

di(E)/g2di(R) on σ in the random case and in random

case with excluded volume (amorphous case too). Thecase of crystals again represents a puzzle.

As it was already pointed out, behavior of PDF in crys-tals is closely related to the behavior of the error termin Gauss’s circle problem that remains under investiga-tion for more than 150 years. It is interesting that thisbehavior that was (almost26) an abstract mathematicalproblem can be measured, in principle, in a real physicalexperiment.

It is clear that the size of oscillations in Gdi(r) is de-termined by the size of fluctuations in the number ofpoints near the surface of d-dimensional sphere of radiusr. That is also true with respect to the behavior of h(r)in crystals, since points deep inside of the sphere are notsubject to fluctuations. However, it is hard to say thatpoints that are fixed on a grid participate in any kindof fluctuations. Thus the origin of these fluctuations, incase the crystals, lies in geometrical compatibility of thesurface of the circle/sphere with its non-zero curvatureand geometry of the lattice whose edges/faces have zerocurvature.

H. The Role of the Spherical Geometry.

Spherical geometry is basically imbedded into the def-inition of PDF due to its connection with diffraction ex-periments. However, in order to illustrate the role ofspherical geometry, it is also possible to define triangu-lar density ρtri(r) or square density ρsqr(r) (and so on...) by counting the number of points inside triangularor square annuluses that are shown on Fig.19. Here weassume that PDF is defined through bins. The “radius”of the triangular density or square density can be defined

FIG. 19: Triangular or square PDFs can be defined throughtriangular or square densities, i.e. by counting the number ofpoints inside triangular or square annuluses and dividing thisnumber by the area of the annulus.

as the radius of the circumference to which this squareor triangle is inscribed.

It is almost obvious from Fig.19 that square densityon the square lattice basically turns 2d problem into 1dproblem. Thus PDF defined according to (7), as G(r) =√

r[ρsqr(r) − ρo], will diverge as r increases. In orderto avoid divergence PDF should be defined, as in 1d, asG(r) = [ρsqr(r) − ρo]. It is not so evident, but it is alsotrue with respect to triangular-pdf. Figure 20 shows thatthe amplitude of oscillations in [ρtri(r)− ρo] persist as rincreases.

It is easy to see that if points are distributed ran-domly then PDF should always be defined (for any poly-gon) according to (7): G(r) =

√r[ρ(r) − ρo]. On the

contrary, if points form a lattice then for any polygonwith finite number of edges PDF should be defined asG(r) = [ρsqr(r) − ρo]. But if the number of edges be-comes infinite (circle) then PDF for the lattices shouldbe defined in the same way as it is defined for the randomdistribution of points.

Thus it is the transition from a finite number of edgesto the infinite one–transition from zero curvature of theedges to non-zero curvature of the circle’s circumferencethat makes random and ordered distributions of pointssomewhat equivalent. Appearance of this equivalence

15

0 5 10 15 20

-1

0

1

2

3

80 85 90 95 100

-1

0

1

2

3

ρ tri(r

) -

ρ ο

480 485 490 495 500

Distance r/a

-1

0

1

2

3

FIG. 20: Dependence of ρtri−ρo on distance r/a for triangularPDF on square lattice. Since the amplitudes of oscillationpersist as distance increases the figure demonstrates that theuse of the polygons (any polygon can be divided into triangles)instead of the circle changes the way in which PDF should bedefined.

changes the way in which PDF should be defined. Thusthe role of the spherical geometry (non-zero curvature)is extremely important for the properties of PDF.

VI. PERFECT S(q)

The quality of the measured scattering intensity and,as follows from (1), of experimental PDF is limited by thefinite instrumental resolution. However, one may want tocalculate “perfect” scattering intensity, as it would be ifinstrumental resolution would be infinitely good. It canbe done by performing inverse Fourier transformation ofperfect PDF that can be calculated if lattice parametersand properties of atomic vibrations are known.

F (q) =∫ Rmax

0

G(r) sin(qr)dr (43)

But, in the case of crystals, PDF does not decay at largedistances and it becomes unclear at what maximum valueof Rmax integration should be terminated.

Figure 21 shows the reduced scattering intensity F (q)calculated from PDF function for the lattice parameters

-100

0

100

200

-200

0

200

400

-100

0

100

200

0 10 20 30 40 50 60

Momentum Transfer q, Å-1

-200

0

200

400Red

uced

Sca

tterin

g In

tens

ity F

(q) A) L = 100 Å, σ = 0.05 Å

B) L = 200 Å, σ = 0.05 Å

C) L = 100 Å, σ = 0.1 Å

D) L = 200 Å, σ = 0.1 Å

FIG. 21: Reduced scattering intensity obtained by Fouriertransformation from PDF calculated for the values of latticeparameters similar to those for the Ni crystal at 20K and300K: a ' 3.53A, σ20 = 0.05Aand σ300 = 0.1A.

of the Ni crystal. It was assumed that atoms of Ni ar-ranged in the perfect FCC lattice and that σ is the samefor all atomic pairs.

The panel A) shows F (q) calculated for the value ofσ = 0.05 A that is close to the value of σ for Ni at 20K.Integration was terminated at the value Rmax = 100 A.The panel B) shows F (q) calculated for the same valueof σ, but integration was terminated at Rmax = 200 A.Panels C) and D) both were calculated for the value ofσ = 0.1 A which is close to the value of σ for Ni atroom temperature. The integration was terminated atRmax = 100 A for panel C) and at Rmax = 200 A forpanel D).

Thus we see that although PDF does not decay, F (q)always decays. Moreover Qmax at which occurs decayof F (q) is determined by σ and not by Rmax. However,Rmax affects the amplitude of the peaks: compare ver-tical scales on the panels A) with B) and C) with D).This behavior could be easily understood. The width ofthe smallest feature in PDF is given by σ. Thus if q isthat big, that sin (qr) makes full oscillation on the length∼ 2σ, contribution of any feature in PDF to F (q) wouldalmost vanish. Thus Qmax2σ ∼ 2π and Qmax ∼ π/σ.Thus increase in Rmax does not lead to the increase ofQmax, but develops the structure of F (q) in the fixedrange of q between zero and Qmax.

This note is important in the sense that sometimes it isassumed that an increase of Qmax in experimental mea-surements would lead to a better quality of PDF. Thisassumption, in general, is wrong. If Qmax ∼ π/σ then inorder to obtain high quality PDF it is more important toimprove instrumental resolution in the range q < Qmax

than increase Qmax.

16

VII. CONCLUSION

The PDF with respect to particular atom can not bemeasured by any presently known technique. In any ex-perimental measurement, the averaging which places allat the origin is automatically performed. Thus, the pairdistribution function calculated in assumption of infiniteinstrumental resolution only decays at large distances ifdifferent atoms at the origin have different atomic envi-ronments. Therefore for perfect crystals the pair distri-bution function does not decay. However even for per-fect crystals the experimental PDF obtained from thediffracted scattering intensity will eventually decay atlarge distances due to the finite instrumental resolution.This means that the measured rate of decay of the mea-sured pair distribution function for very good crystalscould be used to test instrumental resolution. The decayof the PDF at large distances can also be used to in-vestigate the sizes of nano-crystals and strained regions.The persistence of fluctuations in the pair distributionfunction has a connection to a value of the rms fluctua-tions for the Gauss circle problem in d-dimensions, thusproviding a link between ordered and disordered distri-butions of sites. From my point of view it is importantto have better understanding of the origin of this connec-tion. Thus, this problem requires further investigation.However, I do not believe that we (by ourselves) wouldable to move significantly forward with this problem dueto the absence on sufficient knowledge in the area of thelattice point theory.

VIII. ACKNOWLEDGEMENTS

Coordinates of atoms were provided to us by G.T.Barkema.

APPENDIX A: RANDOM CASE IND-DIMENSIONS

1. Ensemble averaging.

Now we consider the case of completely random mediawhen points are randomly distributed in space. In par-ticularly it means that two points can be very close toeach other-the situation that can not occur with atomsin crystals or glasses. We also assume that σij = σ forevery pair of sites i and j. In this case it is easy to makederivations for the general case of d-dimensions. PDFfunction is defined in d-dimensional space as (9).

Gdi(r) = rd−12 [ρdi(r)− ρo]. (A1)

The origin of exponent of r will be clear from the follow-ing. The area of the surface of d-dimensional sphere ofradius r can be written as Sd(r) = Ωdr

d−1, where Ωd isthe total solid angle. Thus Ω1 = 1, Ω2 = 2π, Ω3 = 4π.

Let assume that space is divided into very small boxes.The volume of ith box is dVi = ddri = dΩdir

d−1i dri. The

number of particles in the ith box can fluctuate, but theaverage (averaging over different distributions) numberof sites in the ith box is dni = ρodVi = ρodSd(ri)dri =ρodΩdir

d−1i dri. The usual relation (15) is applied to the

average fluctuation for the number of sites in the box:[dni − dni]2 = [(dni)2 − (dni)2] = dni. In the followingwhile calculating ρdi(r), Gdi(r), and G2

di(r) with respectto a particular site we will drop indexes d and i to sim-plify the notations. Thus for the given distribution ofsites the radial density with respect to particular site canbe written as:

ρ(r) =1

Sd(r)

∑i

exp[− (r−ri)2

2σ2i

]√

2πσ2dni. (A2)

Ensemble averaging of (A2) leads to

ρ(r) =∫ ∞

0

exp[− (r−ri)2

2σ2i

]√

2πσ2

ρ(~ri)dΩdird−1i dri

Ωdrd−1.

Since ρ(~ri) = ρo there is no angular dependence and in-tegration over the angles cancels Ωd in denominator. Forsmall σ the contribution to the integral comes only fromthe region ri

∼= r. Thus we conclude that ρ(r) ∼= ρo.Further:

G2(r) = rd−1[ρ(r)− ρo]2 = rd−1[ρ2(r)− ρ2o]. (A3)

In order to calculate ρ2(r) it is necessary to calculatednidnj :

ρ(r)2 =∑i,j

exp[− (r−ri)2

2σ2i

]√

2πσ2

exp[− (r−rj)2

2σ2j

]√

2πσ2

dnidnj

S2d(r)

. (A4)

In order to calculate dnidnj we note that:

dnidnj ≡ [(dni)2 − (dni)2]δij + dni dnj ,

where the term is square brackets is equal to dni = ρodVi.If i 6= j then the summations over i and j in (A4) can beperformed separately leading to ρ2

o. Thus we can write:

ρ(r)2 =ρo

Sd(r)

∫i

exp[− (r−ri)2

σ2 ]2πσ2

dΩdird−1i dri

Ωdrd−1+ ρ2

o. (A5)

If the difference between rdi and rd in the integrand is

ignored, integration leads to

ρ(r)2 =1

2√

πSd(r)ρo

σ+ ρ2

o.

Finally from (A3) (Sd(r) = Ωdrd−1) we conclude that

g2di(R) ≡ G2

di(r)σ

ρo=

12√

πΩd. (A6)

In the last expression we introduced index i again to un-derline that G2

di(r) was calculated with respect to a par-ticular site.

17

2. Integral approach.

It is useful to consider also another approach for cal-culation of G2

di(r). In the following, for simplicity ofnotations, we drop out index d. Instead of averagingperformed over different distribution of sites one can cal-culate, for a particular distribution, the quantity:

< G2i (r) >=

1R2 −R1

∫ R2

R1

G2i (r)dr =

I

R2 −R1. (A7)

In the following we explicitly show that both ap-proaches lead to the same answer in case of random dis-tribution of sites. It follows from (7,9,A1) that:

I =∫ R2

R1

rd−1[ρ2i (r)− 2ρi(r)ρo + ρ2

o]dr. (A8)

Further we introduce:

I1 =∫ R2

R1

rd−1ρ2i (r)dr, (A9)

I2 =∫ R2

R1

rd−12ρi(r)ρodr (A10)

and

I3 =∫ R2

R1

rd−1ρ2odr. (A11)

Obviously that:

I3 =∫ R2

R1

rd−1ρ2odr = (A12)

ρo

ΩdΩd

d[Rd

2 −Rd1]ρo =

ρo

Ωd< NR2

R1>, (A13)

where < NR2R1

> is an average number of particles in thespherical annulus between R1 and R2.

For I2 using expression (8) for ρi(r), in assumptionσij = σ we write:

I2 =∫ R2

R1

rd−12ρ(r)ρodr (A14)

=2ρo

Ωd

∫ R2

R1

Ωdrd−1 1

Ωdrd−1

∑j 6=i

exp[− (r−rij)2

2σ2 ]√

2πσ2dr

=2ρo

Ωd

∑j 6=i

∫ R2

R1

exp[− (r−rij)2

2σ2 ]√

2πσ2dr

∼=2ρo

Ωd

∑j(R1<rij<R2)

1 ' 2ρo

Ωd< NR2

R1>

It was assumed during the last derivation that (∼= sign)σ << R2 − R1 so that basically only sites inside of the

annulus contribute to the value of the integral. This as-sumption is also used everywhere in the further deriva-tions. We used the sign ' since for the particular dis-tribution of sites the number of sites inside the sphericalannulus can deviates from its average value < NR2

R1>

with standard deviation√

< NR2R1

>.For I1 from (A9,8) we have:

I1 = (A15)∑j 6=i

∑k 6=i

∫ R2

R1

exp[− (r−rij)2

2σ2 ]

Ωd

√2πσ2

exp[− (r−rik)2

2σ2 ]

Ωd

√2πσ2

dr

rd−1.

This expression can be split into the two sums I1 = I11 +I12. In the first sum I11: j = k while in the second I12:j 6= k. Thus:

I11 = (A16)

12√

πΩ2dσ

∑j 6=i

∫ R2

R1

exp[− (r−rij)2

σ2 ]√

πσ2

dr

rd−1

∼=1

2√

πΩ2dσ

∑j(R1<rij<R2)

1rd−1ij

' 12√

πΩ2dσ

∫ R2

R1

1rd−1ij

Ωdrd−1ij ρodrij

=1

2√

πΩd

ρo

σ(R2 −R1).

From (A15) follows that expression for I12 can be writ-ten as:

I12 =∑

j

∑k 6=j

1√4πσ2

exp [− (rij − rik)2

4σ2] · (A17)

· 1Ω2

d

∫ R2

R1

1rd−1

1√πσ2

exp [−(r − rij+rik

2 )2

σ2]dr

The integral in the last expression has non-zero value ifr ' (rij + rik)/2 ± σ. Since σ is small (A17) can berewritten as:

I12∼=

1Ω2

d

∑j(R1<rij<R2)

∑k 6=j

1√4πσ2

· (A18)

· exp [− (rij − rik)2

4σ2] · [ 2

rij + rik]d−1.

Further we substitute summation over k by integrationassuming mean distribution of sites:

I12∼=

∑j(R1<rij<R2)

∫ R2

R1

exp [− (rij−rik)2

4σ2 ]√

4πσ2· (A19)

· 1Ω2

d

[2

rij + rik]d−1 · Ωdr

d−1ik ρodrik

∼=ρo

Ωd

∑j(R1<rij<R2)

1 ' ρo

Ωd< NR2

R1>

18

Since I1 = I11 + I12:

I1 '1

2√

πΩd

ρo

σ(R2 −R1) +

1Ωd

ρo < NR2R1

> . (A20)

Finally for I = I1 − I2 + I3 we write:

I ' 12√

πΩd

ρo

σ (R2 −R1) (A21)

+ ρo

Ωd< NR2

R1> − 2ρo

Ωd< NR2

R1> + ρo

Ωd< NR2

R1>

= 12√

πΩd

ρo

σ (R2 −R1).

Note that in expression above the value of the < NR2R1

>

is determine with precision√

< NR2R1

> (that is the origin

of the ' sign) and that all terms in which < NR2R1

> ispresent directly cancel each other. It is also indirectlypresent in remaining term (see (A16)), however it doesnot affect domination of the main value of the remainingterm.

Thus for g2d(R) we have:

g2d(R) =< G2

di(r) >σ

ρo

∼=1

2√

πΩd, (A22)

which is the same expression (A6) that was obtained fromensemble averaging.

APPENDIX B: AMORPHOUS CASE IND-DIMENSIONS. INTEGRAL APPROACH

1. Difference with the random case from theintegral approach. Detailed derivation.

The derivation for the random case with forbidden vol-ume goes in the same way as for completely random case.The only difference comes from the fact that when thereis forbidden volume around every point expression (A19)should be modified into:

I12∼=

∑j(R1<rij<R2)

∫ R2

R1

exp [− (rij−rik)2

4σ2 ]√

4πσ2· (B1)

· 1Ω2

d

[2

rij + rik]d−1 · (Ωdr

d−1ik − γλd−1)ρodrik

These change comes from the fact that if, in amorphouscase, there is a point j at a distance rij then point kshould be placed in such a way that |rij − rik| ∼ σ.Otherwise exponent in expression (B2) will vanish. Butin amorphous case not the whole volume Ωdr

d−1ij drij is

available for placing point k. There is forbidden vol-ume around the site j: Vf

∼= γλ2drij , where λ can beconsidered as the distance between the nearest neigh-bors and γ is the coefficient that depends on the struc-ture. Thus, see Fig.8, effectively Ωdr

2ikdrik turns into

(Ωdrd−1ik − γλd−1)drik. Further (compare with (A19):

I12 'ρo

Ωd< NR2

R1> (B2)

−∑

R1<rij<R2

∫ R2

R1

exp [− (rij−rik)2

4σ2 ]

Ω2d

√4πσ2

2d−1 · γλd−1ρodrik

(rij + rik)d−1

' ρo

Ωd< NR2

R1> −γλd−1ρo

Ω2d

∑R1<rij<R2

1rd−1ij

Using mean density approximation, we again switch fromthe sum to the integral:

I12 'ρo

Ωd< NR2

R1> (B3)

−γλd−1ρo

Ω2d

∫ R2

R1

1rd−1ij

Ωdrd−1ij ρodrij

' ρo

Ωd< NR2

R1> −γλd−1ρ2

o

Ωd(R2 −R1)

Thus in amorphous case for I = I1 − I2 + I3 we have:

I ' [1

2√

πΩd

ρo

σ− γλd−1ρ2

o

Ωd](R2 −R1) (B4)

So that for < G2di(r) > we get:

< G2(r) >∼= 12√

πΩd

ρo

σ [1− 2√

πγλd−1ρoσ]. (B5)

Since ρo = 1/λd:

g2di(F ) =< G2

di(r) >σ

ρo

∼=1

2√

πΩd[1− 2

√πγ

σ

λ]. (B6)

APPENDIX C: 1D CRYSTAL

From definition of radial density in 1d (8) follows that:

ρ(r) =1√

2πσ2

∞∑n=−∞

exp [− (r − na)2

2σ2], (C1)

while from definition of PDF in 1d (7):

G2(r) = ρ2(r)− 2ρoρ(r) + ρ2o. (C2)

Due to periodicity of the lattice, in order to find< G2(r) >, it is enough to perform integration (10) overany interval of length a. Because of the structure of ex-pression (C1) for ρ(r) and its physical meaning the inte-gral of ρ(r) over any interval of length a should be equalto ρoa = 1, i.e. < ρ(r) >= ρo. Thus, for any R, we canwrite: ∫ R+a

R

∞∑n=−∞

exp [− (r−na)2

2σ2 ]√

2πσ2dr = 1, (C3)

19

while < G2(r) >=< ρ2(r) > −ρ2o. From (C1) follows

that:

ρ2(r) =∞∑

n=−∞

∞∑m=−∞

exp [− (r−na)2+(r−ma)2

2σ2 ]2πσ2

. (C4)

Since

(r − na)2 + (r −ma)2 = 2[r − (n + m)a2

]2 +(n−m)2a2

2,

expression for ρ2(r) can be rewritten. If i = n −m andj = n + m then:

ρ2(r) =∞∑

i=−∞

exp [− i2a2

4σ2 ]2√

πσ

∑j

exp [− (r− ja2 )2

σ2 ]√

πσ2(C5)

In the terms that arise from the last expression i and jshould be both even or odd. Thus for < ρ2(r) > we get

(compare with (C3)):

1a

∫ R+a

R

ρ2(r)dr =∞∑

i=−∞

exp [− i2a2

4σ2 ]2√

πσa(C6)

From it follows that:

< G2(r) >=1

2√

π

ρo

σ[1− 2

√π

σ

a] + (C7)

21

2√

π

ρo

σ

∞∑n=1

exp [−n2a2

4σ2]

Finally:

g21L(σ)

g21R(σ)

= [1− 2√

πσ

a] + 2

∞∑n=1

exp [−n2a2

4σ2] (C8)

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