- 1 . 0 ? 1 - : AL | OF L ORNL P 1396 : I . . . T - . . p . .. S . : ... .. . . SDMI - i ! So 1.. 240 |1:25 || 14 | 15 MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS - 1963 LEGAL NOTICE This report was prepared as an account of Government sponsored work. Neither the Š United States, nor the Commission, nor any i person acting on behalf of the Commission: A. Makes any warranty or representa- tion, expressed or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any information, appa- ratus, method, or process disclosed in this report may not infringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report. As used in the above, "person acting on behalf of the Commission”includes any em- ployee or contractor of the Commission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any information pursuant to his employ- ment or contract with the Commission, or his employment with such contractor. 1 NOTE: This 18 a draft of a paper being submitted for publication. Contents of this paper bbould not be quoted or referred to without permission of the authors ORNr-P-1394 cai F-6-56-(1,79-/ ? ORNI-AIC - OPSICIAL . . hand A NOTE ON THE DETERMINATION OF A METASTABLE MISCIBILITY GAP FROM INTEGRATED SMALL-ANGLE X-RAY SCATTERING DATA* R. W. Hendricks and B. 8. Borie Metals anů Ceramics Division, Oak Ridge National laboratory Oak Ridge, Tennessee 37831 --LEGAL NOTICE - TN. report nu prepared u n account of Govonent sponsored wort. Wellber eine Uwind Bath, mor the Commission, for my pornoo acting as bolall of the Coa rivolon: A. Makes may warruly or reprenution, expressed or implied, wd respect do deserto racy, completeness, or whalsene of the laborator contalo u ao mort, or winner ol any taformation, panou, molhad, or procus dixclound in wompor may not infringt primuly owned roots; or 8. AINDI nay hlabluues with mospect to be w of, or lor damascus roowing from the un of uw lalorestan, apoantu, Bachod, or procou discloud in W. zoport. A. dua abon, "perios kthyos balol the coa mission" include, my on. ployo or moinicior of the C aulon, or plogue of such contractor, to do onni mi Juca enploys or contractor of the Coaalsolda, or onployn of such contractor proper dionbinates, or provides auto, may takorralton purwal ho No esployant or contract with the cornoslou, or We employment mus such contractor, (To be published by Gordon and Breach in the Proceedings of the Conference on Small-Angle X-ray Scattering to be held in Syracuse, New York, on June 2426, 1965), PATENT CLEARANCE OBTAINED. RELEASE TQ THE PUBLIC IS APPROVED. PROCEDURES ARE ON FILE IN THE RECEIVING SECTION, “Research sponsored by the U. 8. Atomic Energy Commission wnder contract with the Union Carbide Corporation, ONNI - AEC - OFFICIAL poslao NOTE: This is a draft of a paper being subuitted for publication. Contents of this paper should not be quoted or referred to without permission of the authors 'ORNI - AEC - OFFICIAL OININEC - OFFICIAL A NOTE ON THE DETERMINATION OF A METASTABLE MISCIBILITY GAP FROM INTEGRATED SMALL-ANGLE X-RAY SCATTERING DATA* R. W. Hendricks and B. s. Borle Metals and Ceramics Division, Oak Ridge National laboratory Oak Ridge, Tennessee 37831 (To be published by Gordon and Breach to the Proceedings of the Conference on Small-Angle X-ray Scattering to be held in Syracuse, New York, on June 2426, 1965). * Research sponsored by the U. 8. Atomic Energy Commission under contract with the Union Carbide Corporation. ORNI - AEC - OFFICIAL ORNI - AC - OSSI Paper to be presented at the Conference on Small-Angle X-ray Ecattering to be held in Syracuse, New York, June 2426, 1965 OINI-MIC-OISICIAL ORNI - AIC - OINICIAL A NOTE ON THE DETERMINATION OF A METASTABLE MISCIBILITY GAP FROM INTEGRATED SMALL-ANGLE X-RAY SCATTERING DATA* R. W. Hendricks and B. S. Borie Metalo and Ceramica Division, Oak Ridge National Laboratory Oak Ridge, Tennessee 37831 ADSTRACT The Gerold and Cowley theories of integrated small-angle x-ray scattering data appear, at first sight, to te lo disagreement regarding their relationship to the phase boundaries of the metastable miscibility gap which is believed to describe the formation of Guinier- Preston zones in quenched Al-Ag, Al-2n, and NaCl-AgCl soud solutione. A new derivation, based on the two-phase model of the zones, shows that the Cowley description of that portion of the total intensity which is sensitive to the local atonic arrangements consists of four physically distinguishable parts, three of which are not observable in most small-angle scattering experiments. On eliminating these terms, the remaining intensity 18 related to the composition of the metastable miscibility gap, but the relationship differs somewbat from the one derived by Gerold. The differences between the two theories are examined in detail, and the metastable miscibility gaf Por querched Al-Ag alloys 18 recomputed. Research sponsored by the V.8. Atomic Energy Commission under contract with the Union Carbide Corporation. ORNI-AEC - OSNICIAL ORNI - AIC -01 I. INTRODUCTION OINI - AIC - OINICIAL ORNI - ASC - ONLICIAL The earliest stages of the decomposition of quenched Al-A8, Al-20, and Haci-AgCl solid solutions are characterized by the formation of small, mpherical, solute-rich regime known as Guinier-Preston zones. The extensive research on G.P. zone formation has been reviewed by Kardy and Heal, . Guinier, 2 and Kelly and Nicholson, and will not be repeated here. Ratber, it do the object of this peper to consider one recent and particularly significant contribution to our understanding of G.P. zone formation - that of determining the phase boundaries of the metastable miscibility gap by the measurement of the integrated small-angle x-ray intensity, to point out an importent criticism of the present interpretation of the method, and to attempt to resolve the apparent conflict. II. MEASUREMENT OF THE METASTABLE MISCIBILITY GAP Based on electrical conductivity measuremente in quenched Al-Ag alloys, Borellus and Larsson* concluded that the formation and reversion of Guinier- Preston zowes was controlled by a metastable miscibility gap. Mors recently Gerold' was shown that the phase boundaries of such a metastable miscibility gap may be calculated from integrated small-angle x-ray scattering data. To date, this theory has been applied to three solid solutions known to form spherical Guinier-Preston zones on quenching; Al-Zn, 5,6 Al-A8, *,7 and NaCl-A8C2." In order to relate the integrated small-angle x-ray intensity to the metastable phase diagram, it io riccessary to absime some model for the structure of the solid solution. Two different models have been suggested, both of which predict the observed shape of the disrube Intenotty distribution. The first model, initially proposed by Quinier,' considers the decaposed solid solution to consist or zones or composition og distributed with some OINI-MIC-Orfirial ORMI - AIC - OilICIAL ORX-A8C - ONNICIAL OINIAIC - ONNICIAL degree of order in a uniform matrix of composition ma, 16 depicted schematically in 118. 1. In this model, the (coberent) zones may be considered to be a second plase in petastable thermodynamic equilibrium with the surrounding matrix. Walker and Guinierio proposed a second model in which the zones, considered to be spherical regiono enriched in alloy atons surrounded by a concentric region impoverished in alloy atome, vere diotributed at random througbout a uniform undecomposed matrix. Gerolds nas argued convincingly in favor of the f1røt model, and that interpretation will be the one used in the following discussian. Using small-angle x-ray scattering theory, it has been shown that the Integrated intensity may be expressed by Que morf Ber(n)an - S 602 odav where I' (n) = scattered intensity (1n electron units per alcm), 26ino ☆ = 261ne - wave 1 incident x-irradiation * atomic volume, * volume irradiated, - local electronic density, and ei - mean electronic density. Using the model of 718. I to describe the electronic density of the decomposed bolid solution, Gerolds has obown that the right-hand side of Eq. (1) may be integrated to give OINTAIC - OISICIAL (2) IVIDIO-DIV-IKIO g plmi - A) (A - 2) (2A - 83)2/& where p to the fraction of the alloy segregated into zones, f, and f, are the atomic scattering factory of A and B atoms, respectively, ORNI - ACC-OSICIAL and my, ma, and we are the composition (10 mole fraction of A-atoms) of the zones, the matrix, and the unde composed solid solution, respectively. Equation (2) suggests that the integrated intensity depends on the way in which the atoms are distributed in the segregated solid solution. Therefore, if the zones are really a thermodynam..cally metastable phase, me and ma are independent of me and by measuring as a tinction of the overal. composition A 10 18 possible to Golve a pair o? Eqs. (2) simultaneously to recover the composition of the metastable phase boundary. o perioada (I) scatterod by an alloy in any state of clustering or short range order may be expressed in terms of the intensity due to the fundamental reflections (Ir), which is not dependent on the way in which the atama arc arranged 10 solid solution, plus a term known as the order intensity (I.) vnich 16 dependent on the way in which the atoms are distributed. Mathematically, Info - Ig * to gue K. - (0,* %)*9.20* 2011 (28+ ngen) + Home (A - B)2 Lamm® (3) como fue Zemer OINI -A1C - Ovnicima Ohn-All-OSICIAL where I = Intensity at X (10 electron units) Amin = vector from atom at siie m to atom ut site a [ + habe + haba) ORNI - AIC - OSPICIAL a emon = 1 - saimn. In Eq. (3) PAR 18 the probability of finding an A-utom at the lattice arter first having found a B-aton at the origin. The vectors bu, ba, and bg are reciprocal to half the crystallographic axes 21, 22, and 13, and are twice the usual reciprocal lattice vectors. The subscript notation used here is somewhat confusing in that the indices (m n) used in the first summation denote the m and oth atom, respectively, while the indices (lma) used in the second summation denote the 21, 82, or 23 component of the vector i'rom the mtb atom ab origin to the oth atom. The conversion from double to single sums is adequately discussed in the literature 12-14 and will not be repeated here. Because the fundamental intensity is independent of the way in which the atoms are arranged in the solid solution, any small- angle x-ray scattering from small, coherent clusters of solute atoms (G.P. zones) must be contained in the order intensity expression. Walker and Guinieriº and Freise, Kelley, and Nicholson 15 have used a Fourier inversion of the structure sensitive term of Eq. (3) to obtain the probability of finding solute atom pairs as a function of pair separation, and have interpreted the results in terms of the denuded zone model described earlier. In order to coupare the above description of the small-angle x-ray scattering with Gerold's model, OINT-ABC - OFFICIAL ORNI - AIC - ON we integrate the order intensity over one repeat volume in reciprocal ORNI - AEC - OFFICIAL ORNI -NIC-ONSICIAL srace to find ar per NT J J J ' Io(hehighg, an dhadhg bomth m (fA - fb)2 ; } } rem VJ JJ Le bune < < Imm 'ahdh dhg. (4) Integrating the right-hand side term by term, and noting that for a cubic crystal the a-series has cosine series, Eq. (4) may be rewritten ag ÀV JIJI (b2babig)dh, dhadhg. --- mam AB. JJ Japoodhdh dhg (5) By the previous definition, Qooo may be seen to be unity since it is impossible to find an A-aton and a B-atom on the same lattice site. The left-hand side of Eq. (4) is simply the value of the order intensity averaged over the volume of one unit cell in reciprocal space. Comparing Eqs. (1) and (3), we note that the variables h in Gerold's notation and hihahg in Cowley's notation differ significantly. Remembering the definition of the diffraction vector, we may find the following relationship \KI = 21| (habz + haba + baba) 1 = 28 - s01 (6) ORNI - ASC - OFFICIAL ORNI -AIC - OFFICIAL - - - - - . . - . 1YIDISJO-DIY-IXIO where s. and 8 are unit vectors la the direction of the incident and diffracted x-rays. Defining b' - (2+ niß + b3)}, Eq. (6) reduces to OEMI - All - Oi!ICIAL bb (7 Converting the integral of the order intensity in Eq. (5) to spherical coordinates, we fir.d (f. - 8,12 | n2I' (n)an = B V (8) where it is defined by TH3 = 1 and insures that the integration 16 performed over the same volume in spherical coordinates as in rectarimunr coordinates. The factor of 2 arises in the following way. The order intensity (I) in Eq. (4) has the periodicity of the reciprocal lattice, and hence, for a face-centered cubic alloy the diffuse scattering found near the origin is also found near each reciprocal lattice point. Since the integration of Eq. (5) simply averages the order intensity sound in one reciprocal unit cell, if we introduce the notation that I(n) measures only the scattering about (000), then to account for all the intensity, we must multiply by 2. In must small-angle scattering experiments, I'(n) bas decreased below the noise level for b < (fr - Fa)(- Fale -X65-7086, -)**.9965-1,6-1,* * 6**$- 40-v************** mm 1K•R IK.R IK•R pg ps F iKRro IK Bpm + 2F (F2 - TALL + 2F (F1- Lle pm • Is r.ke-vojtky - 28-w??c-n pg i LondP Pm • 42-7 {[149 - They . 213-w?£4-**** + 2%, ££ (P. - Radek, (11) rm Equation (2) describes the total x-ray Intensity scattered by the assumed model. The first term gives the fundamental reflections, but the remaining terms must be evaluated and rearranged to put them in a more recognizable form. The terms involving the atomic scattering factors , and fy may be evaluated directly. Consider first the terms ORNI - AFC - OFFICIAL ORNI - ALC - OFFICIAL having the form 2F - Fi)e pom. since the atoms have been assumed to be randomly distributed within the precipitate particle, to may be withdrawn from under the double sum and replaced by its average value . But, by the previous definitions, <> =F1, and OINT-AOC - OFFICIAL . .. KOR Pu! = 0. (12) By similar reasoning 1KApr. 2 XX - Fale- Fale **** - 26<>- F3)<>-F) Pr Pr = 0. (13) In Eq. (13), it is to be remembered that the terms p = r do not exist, as it is impossible for the same site to be simultaneously in both the matrix and a precipitate particle. On the other hand, in the second and third double sums of Eq. (11), the terms p = q and r = 8 must be considered. Examining the double sum over zone sites only, we find EX (? – Fj), - Pude * * - 2218 - Pa)lfe - Fa) Pa +2% 14, -23) - Bebe pa bromel piq = P{ - 28 >F'+ Fi = Pag (1 – ma)(fA – Egle (14), OKML - AEC - OFFICIAL where P 18 the total number of lattice sites in the zones. The same reasoning as for Eq. (12) has been used to show that the terms p ý g do not give a contribution to the intensity. The identity <2> = myra + (1 - mi)pa bas been substituted to give the final expression. A similar expression may be derived for the matrix-matrix double sum. Combining all of these results, the total diffracted intensity becomes ORNI - AIC - UPSICIAL - ALL man + Pmi (1 - mi)(f2 - )2 + (N - P)m2(1 - m2) (f-fp)2 liiKRS STIK Epr (FG-F.)2 + (F1-F.)2)) Pq 29 + 2(F1-F_)(F2-FR) Pr il 1K•Rom. AK R. + 2F (F1 - F.) t 21. (F, )) ) e (15) pm. rm The last two terms of Eq. (15) may be combined with the line above them by noting that pm P a Dr Pr and 175 -- IKLIM - 1L 1K•R 1K.RU tKobrB + 1 p 1K.RO Lule ret Thus, Eq. (15) reduces to ORAL - AEC - OFFICIAL KOR I s TV ORMI - A10 - OINICIAL + Png (1 – ma)(fA - fp)2 + (n − p)m2(1 – ma)(PA- fple + (79 ****+ (P3) - POZ.X ***** + 2(F;F2 .***Pþr Pq KR (B.) р г The first line of Eq. (16) describes the sharp fundamental reflectione, while the second and third terms are the laue monotonic scattering from the zones and the matrix, respectively. The fourth term is scattering due to the size and interparticle interference effects of the zones and the fifth term 1.8 the analogous description of the matrix interactions. The final term arises from the interaction between the matrix sites and the zone sites. The symmetry of this result with respect to the matrix and the zones should be noted. As the average composition ma varies from zero to unity, the role of the matrix sites (r, s) and the zone sites (p, q) continuously interchange with each other. Near equimolar concentration, where it riakes little sense to speak of zones embedded in a matrix, Eq. (16), etill describes the scattered x-ray intensity so long as there is a phary interface between the two phases. In the situation where it is pbysically meaningful to describe the decomposed solid solution in terms of precipitate particles (or zones) of composition my surrounded by, and coberent with, a matrix of composition m12, Eq. (16) may be simplified further by noting that ORNI - AIC-OFFICIAL 1KR pas 1K BOY a KR ng and Kn ORHI-AC-OSTICIAL we wo 42.. .. OIKI - AIC - ONNICIAL po 1x. Ann 1K.. MT 1X.Bc ko Boliba D et han and 1K Bor P1K Bom om 1K.R 19.442.*-I.**. 1KBpa LL р г Pq Substituting these identities into Eq. (16), and using tbe result that, ir the crystal 18 absumed to be infinite, IK A ** * we find + Ping (1 - mi)(CA fp)2 + (N – P)mz(fx – Pogle * (83– Paja 5% -60% 189 -8) [D* 1K.Box (17) This expression shows that the total diffracted intensi'ty scattered by an array of coherent precipitate particles consists of abarp Bragg reflections, two Laue monotonic terms arising from the (assumed) randam distribution issu .,- OINI - ACC-OINICIAL vijiu TVP119-IV-IXIO of the atoms both in the zone 6 and in the matrix, a terme due to the size, shape, and interparticle interference orrecto of tha zoneo, and a (negative) sharp intensity at the reciprocal lattice points which effectively decreases OIMI-AIC - OIFICIAL the Brage Intensity. IV. COMPARISON WITH FREVIOUS WORK In the discussion of the Cowley representation of the diffune scattering, the order intensity (I) vag derined as that portion of the total intensity which was dependent on the way in which the A and B atcmo vero arranged on the lattice dives, and was shown to be the dirference between the total intensity and the fundamental reflections. It was aloo shown that the integrated order intensity was independent of the atomic arrangements. In comparing the results of the present derivation with Cowley's analysis, 1t 18 necessary to include in the order intensity all of the terms of Eq. (17) except the first, which represents the fundamental reflections. The integrated order intensity for the abolmed model 18 then &-* J {m: (1 - mx)(8A {p)2 + (n − p)mz(1 – ma) (SA - fyl? 13-18, -4n-va.. 211K•R +(Fy - F2) 2Le Pg T IKA? F2)2 )...e by dhadhg - #(1 - )(x - * * * = (1 - =)(x - 2) • E1P3 – Pa2 - – ra)2. (28) VKM - ALL-OUSICIAL DINI - AIC - OISICIAL 16 Using the equation for the conservation of 1-ataus OIXI - ATC - ONNICIAL Mon - Som (- Plana Eq. (17) becomes 8 - A62 - A) KEA fp)2 09) which 18 identical to the integrated order intensity given by the Covley theory. Warn the order intensity computed for the model given here (290. (16) or (17)) la compared directly with the Fourier serdec rcprcscntation given by Cowley (Eq. (3)], we notice that, due to the absumed arrangement of the atoms, the order intensity is separated into four recognizable terms. The integration performed in Eq. (13) sbove that, although cach of these termo depend on the local atomic arrangements, their own 10 dependent only on the average composition from an experimental point of view if any one of these four terme lo unobservable, then the observed value of ll will no longer be independent of the atomic arrangements. An examiantion of Eq. (17) shows that the last term dexcribes a series of sharp reflections which accur at the same locations as the fundamental reflections, and will not be separable from them. In small-angle scattering experimentu, this term vil: be in the incident intensity, and cannot be detected. Albo &csuming reasonable values for the compooltion of the zoncs (az) and the matrix (m2), and for the volume fraction of the zones , it may be shown that roughly one-half of 18 aboociated with the Lave monotonic intensity scattered by the zones and the matrix, while moct of the remainder 16 associated with the teru ardoing fras the size, shape, and distribution of the zoneb. Only a small portion (about one-tenth) URHI-AIC-OINICIAL 17 ORNI - AC - OFFICIAL 16 associated with the negative sharp reflection. The Laue monotonic intensity 16 spread uniformly over the entire repeat volume in reciprocal space, while for particle 61z.es of interest in small.-angle scattering experiments (101 to 10001 radius), it 18 found that the particle-size owwnation describes a broad peak near the center of the unit cell (1n reciprocal spuce) which has decreased below the noise level of the experiment for values of h approximately one-tenth of the distance to the edge of the cell. Thuo, roughly one-half of the total observable dirruse intensity 18 piled up in about one-thousandth or the repeat volume, while the remaining half 16 spread uniformly throughout it. Order of magnitude calculations show that in most small-angle scattering experiments, the Laue monotonic intensity 18 80 weak at any point in reciprocul apace it cannot be resolved from the background nuite of the experiment.* On the bas18 or the be arguments, it may be concluded that, although in theory the integrated structure-sensitive intensity 10 Independent of the local atomic arrangemente, in practice, the observable integrated small-angle x-ray scattering 18 indeed related to the boundaries of the metastable miscibility gap. . To capleto the discubbion, the observable integrated intensity predicted by the present theory must be compared with that predicted by Gerold. Making Budstitutions for F, and Fa, and using the relationship describing the conservation of A-atons, Eq. (18) gives (observed) muz)?(FA p)2 - imz - ma)(m2 - x)(fa - fp)2 (20) *In high-angle diffuse scattering experiments, where the incident intensity iu much greater than in ama ll-angle experients, the Laue ORNI - AIC - OSSICIAL olvidov. diridos monotonic scattering 18 observable. 18 ORNI - AIC - OFFICIAL Comparison with Eq. (2) shows that this result differs from Gerold's expression by the factor (my - ma)/(m2 - m). The reason for the difference between the two equations becomes evident on comparing Eqs. (1) and (16). In Eq. (1) only terms involving the summation (or integration in the continuum approximation) over zone-zone (pa) and matrix-matrix (ro) pairs are included. The matrix-zone pairs are included. The matrix-zone pairs (pr and sq) and the pairs of zero length (p = q and r = 8) which appear in Eq. (16) have been neglected. It is easily proved that if the two double sums in Eq. (16) which involve only matrix-matrix and zone-zone pairs are integrated over .one reciprocal unit cell, the result of Eq. (2) is obta:.ned directly. Such an integration corresponds directly with Eq. (1). Thus, the difference between the final result of Eqs. (2) and (20) 18 due entirely to the omission of the matrix-zone interaction terms. . v. DISCUSSION Using the present derivation, the phase boundary or the metastable miscibility gap in Al-Ag may be recomputed using the data from Bauer and Gerold." The results of such calculations are shown in Fig. 2. It is seen that the shape of the metastable pbase diagram 18 unchanged, but its location is shifted to lower values of my. A similar decrease in zone composition was calculated from Gerold's data for Al-20.5 The present calculation of the composition of the zones (my) and the matrix (m2) 16 still in disagreement with both the results obtained by x-ray and electron microscopy by Freise et al.25 They interpreted their results in terms of the denuded zone model of Walker and Guinier, 20 divisiv. do. ito: ORNL - AEC - OFFICIAL and obtained an equation relating the compositions of the pbasco ORMI - AIC • Tricial to the zone radius. Using small-angle x-ray scattering data, they found that for a quenched 15.6 wt % Al-Ag alloy aged at 125°C, my = 0.33 and ra = 0.03. However, w8106 the mean particle diameter obtained from transmission electron microscopy data from the same sample, they found mı = 0.94 and ma = 0. More recently, Bauer and Gerola 16 have shown that zone radii measured by electron microscopy and small-angle scattering differ by less than 10%, thus bugscoting that the wide variation in zone composition calculated by Preise et al." does not exist. It is not possible at this point to determine which o. their results is more nearly correct, but their lower value (obtained Prom x-ray data) would seem to be indicated. It is interesting to note that Eq. (20) may be solved for ma vithout actually converting the intensity to absolute units because the solution involves taking the ratio of two values of obtained from two diffcrer.t mean compositions. Once having ma, the zone composition my can be obtained only if the data 16 converted to absolute units. Thus, the error in ma is determined primarily by the accuracy with which the relative values of the values of can be determined, while the error 10 12 16 almost directly proportional to the error in the calibration constant which is used to convert the intensity into absolute units. Because of a lack of published data, she accuracy of the zone compositions derived. in Fig. 2 cannot be determined.. OIMI - AIC - ONLICIAL 20 ORAL-ALC - OFICIAL 11121310- DIY - INYO A final result of the present work is related to the use of the integrated small-angle scattering data to convert the observed . intensity to absolute units. Following Cowley, 12 Walker and Guinier, 10 and Freige et al.. 15 both used a Fourier Inversion of Eq. (3) to obtain the probability of finding Ag-Ag pairs of various lengths. In both of these papers, the x-ray intensity was converted to absolute units by noting (Eq. 5) that in this representation the integrated order intensity was a constant which could be evaluated both experimentally, and from theory. The arguments given in this paper suggest that in practice the integrated intensity does not measure the total order intensity, and hence all of their values of PASSAS are too large (except P48543, which must be unity). That such is the case was found by Freise et al. 15 when they also converted their data to absolute units by measuring the intensity of the incident beam with a series of foils, and found that the values of pAB-Ag computed in this way were all considerably lower ..- . - - - - than the previous values. Bauer and Gerold' have suggested that these latter calculations gave values that were too low because the conversion to absolute units was also too low. VI. SUMMARY We bave shown that in solid solutions where the solute atoms are clustered into zones the integrated order intensity described by Cowley consists of four physically distinguishable terms. Each of these terme is dependent on tha way the atoms are distributed between the zones and the matrix, but their sima is not. It has also been shown that at least one, and usually three, of the four terms cannot be observed ORRL - AEC - OFFICIAL ORNLAIC - OFFICIAL 21 14121119-RY-IMO with standard small-angle x-ray scattering apparatue. Thus, the integral of the observed scattering data is indeed related to the metastable miscibility gap, but the relatonship differs from that originally derived by Gerold.5 The reason for the difference is due to the omission of zone-matrix interactions in the original derivation, OXNI - ALC - OFFICIAL and which have been accounted for in the present work. Finally, it is shown that, because some portions of the Cowley order-intensity are not observable, the normallization procedure involving the integrated intensity as used by Walker and Guinier10 and Freise, Kelley, and Nicholson15 is not valid. VII. ACKNOWLEDGEMENTS The authors wish to express their thanks to Dr. C. J. Sparks for his most valuable discussion and comments during the progress of this work. ORHI - AEC - OFFICIAL ORNI - AEC - OFFICIAL REFERENCES ORMAIC-OFFICIAL 1. Hardy, H. K., and Heal, T. J., Progr. Metal Phys. 5, 143 (1954). 2. Guinier, A., Solid State Physics 2, 293 (1959). Kelley, A., and Nicholson, R. B., Progr. Mater. Sci. 10, 151 (1962). Borelius, G., and Larsson, L. E., Arkiv Fysik 11., 137 (1956). Gerold, V., Phys, Status Solidi 1, 37 (1961). Gerold, V., and Schweizer, Wis Z. Metallk, 52, 76 (1961). Bauer, R., and Gerold, V., Acta Met. 10, 637 (1962). Hendricks, R. W., to be published. Guinier, A., J. Phys. Radium 8, 124 (1.942). Walker, C. B., and Guinier, A., Acta Met. 1, 568 (1953). 11. Guinier, A., and Fournet, G., Small Angle Scattering of X-rays (John Wiley and Sons, New York, 1955). 12. Cowley, J. M., J. Appl. Phys. 21, 24 (1950). 13. Warren, B. E., and Averbach, B. L., Modern Research Techniques in Physical Metailurgy (ASM, Cleveland, 1953), pp 95-130. 14. Sparks, C. J., and Borie, B. S., to be published by the AIME in the Proceedings of the Conference on "Local Atomic Arrangements by X-ray Diffraction" held in Chicago on February 15, 1965. 15. Freise, E. J., Kelley, A., and Nicholson, R. B., Acta Met. 9, 250 (1961). 16. Bauer, R., and Gerold, V., Acta Met. 12, 1449 (1964). ORNI-AEC - OFFICIAL ORNI-ACC-OFFICIAL ORNI - AIC-OFFICIAL FIGURE CAPTIONS Fig. 1. (a) A two-dimensional slice through an alloy containing : spherical G.P. zones. (b) The composition of the alloy along the line cc'. Fig. 2. The equilibrium phase diagram for Al-Ag. The metastable miscibility gap found by Bauer and Gerold (see reference 7) is shown by the full line, while the one predicted from their data using the present theory is showa by the dotted line. . .. • - ORNI - AEC - OFFICIAL ORN CALONscin - - - .... ........... . . -....-: -- : * inn o....... . . . . . . . --.... i distance igure o Agatones) (mol fraction of composition • ORNI - ALC - OFFICIAL ORNI - AEC - OFFICIAL ORNI-AIC - OFFICIAL 600 Temperatura, °C ل لللله 20 60 Concentration, AI.% Ag ... ... .. Figure 2 ORNE – AEC -OISICIAL - END . . DATE FILMED 9/13/65 tidak ada saat me n s proposed Mentions like the reader anderen Sie meisies van die kinderen en ondernemin. .