Tracing the origin of the first sharp diffraction peak (FSDP) of sodium metaphosphate glass and melt

Journal of Non-Crystalline Solids 277 (2000) 15±21

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Tracing the origin of the ®rst sharp di€raction peak (FSDP) of sodium metaphosphate glass and melt F. Hajdu Hungarian Academy of Sciences, Institute for Solid State Physics and Optics, Rakoczi ut.12, H-1072 Budapest, Hungary Received 15 October 1999; received in revised form 14 February 2000

Abstract The phenomenon of `®rst sharp di€raction peaks' (FSDP) present in the di€ractograms of certain amorphous solid or liquid systems below the so-called main di€raction peak is studied by the use of Fourier transforms remaining within the di€raction theory of non-crystalline systems. Performing the inverse transformation of the reduced radial distribution functions (RDFs) in separate parts enables us to trace the origin of an FSDP ± or any other features of the di€raction pattern ± to real space. This means that all pieces of information on the (disordered) structure contained by the structure factors are comprised of the pair distribution functions too. The input of extraneous distances, layers, etc. is unnecessary and therefore misleading. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction Several metaphosphate and borate glasses have recently been studied by Herms and coworkers below and above the melting temperatures …Tg ˆ 6 ÿ 800 K† using X-ray and neutron diffraction. By use of a special heatable sampleholder the samples could be studied in both liquid and solid (glassy) states as reported in [1±4]. The intention of the author was to study the structural changes in glass due to higher temperature and melting (i.e. eventual variations of the short-range order (SRO)). This meant also checking a widely accepted view according to which the structure of a glass is the same as that of a frozen liquid. A `frozen liquid' structure should yield a similar di€raction pattern, structure factor, and reduced

E-mail address: [email protected] (F. Hajdu).

radial distribution function (RDF) as the liquid but exhibiting a sharper spectrum of pair distances. The studies of Herms et al. have con®rmed this view for a part of the phosphate specimens but in other cases important changes in the di€raction curves were observed. Especially a ®rst sharp diffraction peak (FSDP) appeared in the X-ray and neutron di€raction patterns of sodium metaphosphate …NaPO3 † around Q ˆ 11 nmÿ1 (Q ˆ 4p sin H=k is the momentum transfer, H the half scattering angle, and k is the wavelength) ± rather unexpectedly ± not in the glassy but in the molten state (NaPO3 glass, and melt at ambient temperature, and at 1020 K, respectively). The phenomenon is not a common feature of chemically relative systems. For example, molten KPO3 and Ca…PO3 †2 exhibit no FSDP. On the other hand, the neutron di€raction pattern of KPO3 glass contains the FSDP thus showing the complexity of the question. The referred authors transformed

0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 0 ) 0 0 2 9 8 - 2

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F. Hajdu / Journal of Non-Crystalline Solids 277 (2000) 15±21

the coherent X-ray intensities into the structure factor S…Q† ˆ Icoh …Q†

.X

xj fj2 …Q†;

where Icoh is the measured and corrected coherent intensity, subscript j refers to the component atomic species, xj the atomic fraction of the jth component, and fj …Q† is its atomic X-ray amplitude. Instead of these amplitudes, neutron diffraction data are related to bj coherent neutron scattering lengths which are independent of Q. Sine Fourier transformation of Q‰S…Q† ÿ 1Š yields the reduced RDF G…r†. From this, the RDF D…r† ˆ rG…r† ‡ 4pr2 q0 , and the dimensionless pair ÿ1 correlation function (PCF) g…r† ˆ …4prq0 † G…r† are easily derived where q0 is the mean atomic density. By studying and comparing the S…Q† and G…r† functions, the referred authors have come to the conclusion that the exceptional di€racting features should be ascribed to special electron density conditions in the medium distance range of NaPO3 melt di€erring from those of the glass and of other studied materials without FSDP. In the past three decades many authors have also found and studied FSDPs (often called pre-peaks) in very di€erent branches of amorphography such as aqueous electrolyte solutions, semiconductors, glassy metals, and pure and mixed molecular glasses as well. As examples of such works we refer to papers of Jergel and Mrafko [6] on the in¯uence of heat treatment and relaxation on some amorphous alloys. Gaskell and Wallis [7] claim that a Bragg-re¯ection on `quasilattice planes' should produce the FSDP of silica glass and of most amorphous materials. Elliot studied the e€ect of network-former and modi®er components in glasses (see [8]). Sv ab and colleagues studied am. Ni±Nb alloys with isotope substituting neutron di€raction and have found among the partial diffraction patterns pre-peak (FSDP) as well as negative pre-peak (di€raction minimum), and smooth low range. Their report and interpretation can be read in [9]. Numerous authors have found the cause of the FSDP among the pair distances of the medium range and the conclusions were mostly

bound to the substances under study as ad hoc explanations. The goal of the present work is to show the use of a rather formal but generally applicable method for tracing the rise of the FSDP to the structure of the amorphous system without abandoning the scope of the di€raction theory of disordered condensed materials. 2. Precedent own work The author and colleagues in their former studies on glassy alloys ([10±12]) applied a quantitative modelling method on amorphous structures in which every peak of the reduced RDF is approximated with a series of Gaussian functions P (peaks) Gi …r; Ri ; Ki ; bi † ‡ Gc …r† where subscript i denotes the serial numbers of the peaks, Ri is the ith pair distance, Ki the ith coordination number, bi the rms ¯uctuation of Ri , and Gc …r† is a closing term. In addition, the structure factor could be established in analytical form as the Fourier transform of the above Gaussian expressions: X S…Q† ˆ CKi exp‰ÿb2i Q2 =2Š sin…Ri Q†=Ri Q ‡ Sc …Q†; where notations are the same as above, and Sc …Q† is the transform of the closing term (the main part of the expression is identical to Debye's intensity formula multiplied with a Gaussian damping factor). The primary aim of this approximating procedure was then to obtain all the structural information, which is hidden in the reduced distribution function of the amorphous system, in the form of a set of well-de®ned physical quantities (Ki coordination numbers, Ri pair distances, and bi rms deviations). As a secondary result, the origin of unusual features of the structure factor (and those of the experimental scattering pattern itself), like FSDPs (alias `pre-peaks'), shoulders, and split peaks, could be traced when the single Sk …Q† terms were summed up one by one and the partial sums checked for the presence or absence of the feature in question. The accuracy of the reproduction depended naturally on the goodness of the ana-

F. Hajdu / Journal of Non-Crystalline Solids 277 (2000) 15±21

lytical ®t for the G…r† function as it was shown in [10±12]. 3. Reproducing the X-ray structure factors At present, the former secondary result has become our main goal, so we aimed at detecting the rise of an FSDP of molten Na-metaphosphate. The analytic approximation procedure was omitted and no conjectures were made as to the structures of glassy and molten NaPO3 . A great number of measured intensity curves, structure factors, and RDFs are presented graphically by Herms and coworkers [1±4]. In the present work, however, we could start from detailed DIFF…r† ˆ rG…r† data sets of glassy and molten NaPO3 , and Ca…PO3 †2 kindly communicated by Herms to the present author in [5]. Both r and DIFF(r) values are given to seven decimal places. These data were derived from X-ray measurements made with commercial rotating anode device and with synchrotron radiation.

17

As a ®rst step, the data were transformed by us into G…r† ˆ …1=r†DIFF…r†

…1†

utilizing the ®rst three decimals of the abovementioned data tables. The G…r† functions of glassy and molten NaPO3 , and Ca…PO3 †2 were slightly smoothed in order to be freed from spurious maxima and minima in the 0 < r < 0:1 nm range and divided, after some trials, into four sequential sub-ranges. The number and selection of the sub-ranges were chosen so that the appearance and forming of the FSDP should be clearly shown using only the necessary number of partitions. Fig. 1 shows the division zones of G…r† ÿ s. The r-sites of partition were chosen mathematically so that G…r† have there a local minimum (®rst derivate zero) and possibly a zero value too. The partial G…r† ÿ s were then transformed into partial structure factors. (Not in the sense of partial S…Q† ÿ s of binary chemical systems!)

NaPO3

Fig. 1. Experimental reduced radial distribution curves G…r† of NaPO3 samples in the the glassy (300 K) and the melt (1020 K) states. Their division into four parts.

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F. Hajdu / Journal of Non-Crystalline Solids 277 (2000) 15±21

As it is known, Q‰S…Q† ÿ 1Š can be obtained by the (reverse) sine Fourier transformation from the G…r† function. The used formula can be written as 2 G…r† $ Q‰S…Q† ÿ 1Š: F:T: p

…2†

The sequential maxima of the G…r†, (and RDF…r†, g…r†) functions represent the di€erent interatomic pair distances occuring in the sample substance. The r positions, areas, and widths of these peaks correspond to the lengths, the relative occurrences or coordination numbers, and the length ¯uctuations of the distances, respectively. The inverse Fourier integral of G…r† yields the structure factor in its reduced and weighted forms as Q‰S…Q† ÿ 1Š. The Gk …r† fragments were dealt with as being formally complete partial reduced distribution functions G1 …r†; . . . ; G4 …r† when they are supposed to be accomplished with zeros which means that the kth fragment was virtually extended over the whole Fourier integration range of G…r† by taking G…r† ˆ 0 within the sub-ranges r1 ˆ 0 < r < rk , and rk‡1 < r < rmax . Carrying out the Fourier transformation on all the four functions we get four separate partial structure factors S1 …Q†; . . . ; S4 …Q†. While S…Q† oscillates around the constant ˆ 1, S…Q† ÿ 1 curves oscillate around the zero-axis. In the lowest Q range both S…Q† and S…Q† ÿ 1 take up a relatively very high value because lim P S…Q† ˆ Ki for Q ! 0, where the Ki s are the

3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0 -0,5 -1,0 -1,5 -2,0

3,5

NaPO3 300K

Q.[S(Q)-1]

coordination numbers belonging to each of the observable pair distances. This singularity at Q ˆ 0 is automatically eliminated from the weighted structure factor Q‰S…Q† ÿ 1Š which is, on the other hand, just identical with the integrand of the Fourier integral yielding the G…r† function and vice versa. At the same time, Q as a weighting factor of ‰S…Q† ÿ 1Š reduces the relative height of the FSDP as compared to the other peaks of the structure factor. Considering the pros and cons we have chosen the function Q‰S…Q† ÿ 1Š for showing the results of this work in Figs. 2 and 3 exactly from Q ˆ 0 nmÿ1 to Qmax ˆ 140 nmÿ1 . To visualize the actual rise of the FSDP we draw the curves of the four partial structure factors one after the other, comparing them to the total Q‰S…Q† ÿ 1Š as the ®fth curve in Figs. 2(a) and (b). An even more instructive picture can be obtained in the following way: (i) we take the curve S1 …Q† derived from the ®rst function G1 …r† which is non-zero between r1 and r2 ; (ii) we add the second curve to the ®rst one and draw this partial sum: …1st ‡ 2nd† under (or above) the ®rst one; (iii) we continue this procedure up to the last term S…Q† ˆ S1 …Q† ‡


‡ S4 …Q†. The group of curves clearly shows, when and where the FSDP of the structure factor will be produced upon adding together the subsequent contributions as depicted in Figs. 3(a) and (b).

(a)

3,0

Q.[S(Q)-1]

2,5

1

NaPO3 1020K

2,0

1 2 3 4 5

1,5

2 3 4 5

(b)

1,0 0,5 0,0 -0,5

Q[nm -1] 0

20

40

60

80

100

120

140

-1,0 -1,5 0

20

40

60

80

100

Q[nm -1] 120

140

Fig. 2. Partial and total structure factors of glassy (a), and molten (b) NaPO3 . The partial factors (1±4) and the total factor (5) are shown separately. Ordinates shifted by 0.5 unit.

F. Hajdu / Journal of Non-Crystalline Solids 277 (2000) 15±21

NaPO3 300K

3 2

Q.[S(Q)-1]

2,5 1,5

1 2 3

0

4

20

40

60

80

100

120

1

0,5

2 3 4

-0,5 -1,0

Q[nm -1] 0

1,0 0,0

-1

Q[nm -1]

-1,5 140

(b)

Q.[S(Q)-1]

2,0

1

-2

NaPO3 1020K

3,0

(a)

19

0

20

40

60

80

100

120

140

Fig. 3. Similar to Fig. 2. The partial factors are summed up successively: 1 ˆ 1 partial; 2 ˆ 1 ‡ 2 partials; 3 ˆ 1 ‡ 2 ‡ 3 partials; 4 ˆ 1 ‡ 2 ‡ 3 ‡ 4 partials ˆ total. Ordinates shifted.

1,1

NaPO3

S(Q)

+++300K ___1020K

1,0

0,9

0,8

Q[nm-1]

20

40

60

80

100

120

140

Fig. 4. S…Q† curves of glassy (300 K) and melt (1020 K) NaPO3 .

Fig. 4 shows the S…Q† curves of NaPO3 glass and melt together, giving a clear picture of the presence of an FSDP for the melt and a low shoulder for the glass. The same calculations were performed on the data sets of Ca…PO3 †2 and the absence of FSDP has been reproduced. This probe excluded the risk of an eventual artifact in the case of NaPO3 . Let us notice that the FSDP is more striking in the measured X-ray di€raction pattern than in the S…Q† and Q‰S…Q† ÿ 1Š curves because of simple mathematical reasons. It is a trivial task to re-

produce the measured coherent intensity function from Q‰S…Q† ÿ 1Š when a comparison between the simulated FSDP with the original measured one seems to be desirable. 4. Results Na-metaphosphate exhibits an FSDP about Q ˆ 11 nmÿ1 in the melt state while only a low shoulder in its glassy state. Our simulation has reproduced these experimental facts. The question

20

F. Hajdu / Journal of Non-Crystalline Solids 277 (2000) 15±21

is, where the FSDP takes its origin from. As Figs. 2 and 3 show, no positive maxima are present around 11 nmÿ1 as long as the contribution of the G…r† range 0:5 < r < 0:7 nm is not added. The FSDP appears in the S…Q† structure factor of the molten sample only upon addition of the two contributions from ranges 0:5 nm < r < 0:61 nm, and 0:61 nm < r < 0:7 nm, respectively. The latter range seems to be more e€ective. At the same time and site, only a weak shoulder is reproduced in the S…Q† of the glassy sample. Figs. 3(a) and (b) show the presence of the FSDP in the third and fourth curves which are partial sums of the partial Q‰Si …Q† ÿ 1Š structure factors. From Figs. 2(a) and (b), it can be seen that neither of partials 2nd, 3rd, and 4th show up an FSDP themselves but it is also evident that the relative positions of their ®rst and second maxima are di€erent. Fig. 5 shows that the G…r† peaks of the melt and glass in sub-range 2 are nearly at the same r-position (0.575±0.58 nm), those in sub-range 3 deviate from each other: 0.65 and 0.67 nm for the melt and the glass, respectively. Both G…r† peaks are higher and narrower from the glass than those from the melt, yet the lower and broader peaks of the melt produce the FSDP. This means that it is only the whole ensemble of pair separations which either develops the FSDP or not. FSDP formation cannot be ascribed neither to a unique G…r† maximum at a certain r-site, nor to its pro®le. 0,5 0,4 0,3

NaPO3

G(r)

+++++300K _____1020K

0,2 0,1 0,0 -0,1

2

3

-0,2 -0,3

r[nm] 0,55

0,60

0,65

0,70

0,75

Fig. 5. The ranges of the reduced RDFs G…r† of glassy (300 K) and melt (1020 K) NaPO3 , especially important in causing the FSDP.

5. Conlusions Some general conclusions can be deduced by considering earlier experiences together with the results reported above. (i) The explanation of FSDP should remain within the scope of the di€raction theory of noncrystalline condensed systems (amorphography, RDF analysis). Even a sharp FSDP can be formed by the interference of sinusoidal partial S…Q† waves. (ii) In spite of the striking appearence of an FSDP in the raw di€raction pattern, its in¯uence on G…r† can be less signi®cant. As written above, the relative weight of an FSDP becomes rather small in Q‰S…Q† ÿ 1Š. (iii) The position of the main di€raction peak Q1 is greatly determined by the shortest radial interatomic distance r1 , its area, i.e., its height and width by its occurrence. All the longer distances must contribute with one or more maximum and minimum within the range 0 < Q < Q1 . Very often they cancel each other forming a more or less smooth curve section but it is not a strict law. (iv) Most probably a densely packed monatomic atomic system does not exhibit a FSDP. The contribution of two or more atomic species with di€erent scattering amplitudes creates the possibility for a FSDP and deviations from dense packing (as e.g. a tetrahedral arrangement) give more chance to this phenomenon. Sometimes the X-ray and the neutron di€raction patterns of the same sample di€er from each other as to the existence of a FSDP which appears mostly in the neutron pattern. This phenomenon is understandable considering the fact that natural chemical elements consist of several isotopes with various scattering lengths. (v) Most probably, a FSDP in itself carries no direct information on the structure. The question, whether the presence and absence of the FSDP is due to some well-de®ned strucural changes, or to a mere hazardous combination of scattering contributions, falls beyond the scope of the present study. (vi) Conclusions (i)±(v) may hold for randomly oriented and distributed atomic systems. In molecular glasses and melts intramolecular covalent

F. Hajdu / Journal of Non-Crystalline Solids 277 (2000) 15±21

bonds can sustain second neighbor pair distances very ®rmly which should produce a characteristic main maximum between 0 and Q1 at the site of a FSDP. Yet, this is not a FSDP in the common use of the word. May be, this could be the explanation for the ®rst strong peak of silica glass described and interpreted in [7]. Mixing in (crystalline) Bragg-re¯exion theory and calculation seems unnecessary and therefore false. Such a procedure would be rightful for liquid crystals whose diffraction properties are, of course, di€erent from those of common glasses and liquids. In any case, the modi®ed Fourier integration method reported in this paper can generally be utilized as a check for any structural models and attempts to interpret FSDPs and other uncommon features of amorphous and liquid di€raction patterns. Acknowledgements The author thanks Dr G. Herms (Rostock, BRD) for calling his attention to the exciting experimental results, permitting their reprocessing,

21

and for criticising the ®rst conception of the paper. Thanks are due also to Dr E. Svab (Budapest) for her valuable remarks. The work was supported by the OTKA grant T 029402.

References [1] G. Herms, U. Hoppe, W. Gerike, W. Sakowski, S. Weyer, in: Proceedings of the XVII International Congress on Glass, International Academic Publisher, Beijing, vol. 2, 1995, p. 110. [2] U. Hoppe, G. Herms, W. Gerike, J. Sakowski, J. Phys. Cond. Mat. 8 (1996) 8077. [3] G. Herms, J. Sakowski, W. Gerike, D. Stachel, Z. Naturforsch. 53a (1998) 874. [4] G. Herms, J. Sakowski, W. Gerike, U. Hoppe, D. Stachel, J. Non-Cryst. Solids 232±234 (1998) 427. [5] G. Herms, private communication, 1998. [6] M. Jergel, P. Mrafko, J. Non-Cryst. Solids 85 (1986) 149. [7] P.H. Gaskell, D.J. Wallis, Phys. Rev. Lett. 76 (1996) 66. [8] S.R. Elliot, J. Non-Cryst. Solids 182 (1995) 40. [9] E. Sv ab, Gy. Mesz aros, J. Tak acs, S.N. Ishmaev, S.I. Isakov, I.P. Sadikov, J. Non-Cryst. Solids 144 (1992) 99. [10] F. Hajdu, Phys. Status Solidi A 60 (1980) 365. [11] F. Hajdu, Phys. Status Solidi 61 (1980) 141. [12] E. Sv ab, F. Hajdu, Gy. Mesz aros, Z. Naturforsch. 50a (1995) 1205.