. . TI 1 { : 1 OF 2 ORNL P 1240 . . · C . . I PREFEFFE . . non |1:25 L4 LG MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS -1963 .. . - - - - - - - : : - . . SU VEOMA - - . . LEGAL NOTICE This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any 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. . . . . - ---.. .... 17, . .... --. - - . ...... . .. . . ..... ...... - - - - Vaat- - . ... ... *** ... ... . .. ... .. . . HY AT 1 557 4** WELCLIESEGYES 11 2 * - ORNA -P_4240 NOTE: This is a draft of a paper being submitted for - Alication. Conten of this paper should not be quoted nor referred to without permise. of the ethors. CONF-6502030 ORNI - AEC - OFFICIAL MAY 17 1969 - LEGAL NOTICE - MASTER TW, raport mo prepared in accorat of Government sponsored work. Neither the Vallad stata, sors Counsson, for wy porno selo on badall of the Commisoa: A. Make, nay wuruty oi reprenouatioa, expressed or implied, with respect to the accu- racy, completeness, or unetu.out. of the information coatsland la tlo roport, or that the u. of any information, apparatus, mothod, or proceso disclosed la Wa raport may not larriage prinitaly ortod rights; or B. Asmumos hay Habilities with respect to the uni of, or for damagoj resulting from the uw of any information, apparitu, metood, or process dlaclosed in the roport As und in the above, “por ai acdag oo babait of the Commission" includer my om- ploym or contractor of the Commission, or employee of such contractor, to the oxtont wat toch aaploys or coatructor of the Commission, or employs of much contractor preparos. discontratı, or provides accOna to, say loformation pursuant to als omployumot or contruct milt Commigoloa, or his employment with such contractor. METHODS OF ANALYSIS FOR LOCALLY-ORDERED ARRANGEMENTS AND ATOMIC DISPLACEMENTS BY X-RAY DIFFRACTION C. J. Sparks and B. Borie (TO ? pubz: shed by the AIME in the Proceedings of the Conference on i ocal tomic Arrangements Studied by X-ray Diffraction" held in "?icago on February 15, 1965) PATENT CLEARARCE OBTAINED, RELEASE TO THE PUBLIC IS APPROVED. PROCEDURES ARE ON FILE IN THE RECEIVING SECTION. Metals and Ceramics Division OAK RIDGE NATIONAL LABORATORY Apri... 22. 1955 02KI-her-neririal CONTENTS ORNL - AEC - OFFICIAL INTRODUCTION 'l. Theory 1.1 Introduction 1.2 Order Diffuse Scattering 1.2.1 Special Case of Cubic Solid Solutions 1.3 Atomic-Displacements Diffuse Scattering 1.3.1 Special Case of. Cubic Solid Solutions 1.4 Recovery of Order and Atomic-Displacement Coefficients of the Diffuse Intensity : 1.4.1 Separation of Order and Atanic-Displacement Modulations 1.4.2 Evaluation of Order and Atomic-Displacement Coefficients 1.4.3 Treatment of One- and Two-Dimensional Data 1.5 Application of Theory to Powder 1.6 Intensity Expression for Huang Scattering 2. Experimental Techniques 2.1 Conversion of Intensity Measurements to Absolute Units 2.1.1 Intensity Expression in Absolute Units 2.1.2 Measurement of Power in the Incident Beam 2.2 Corrections for Background Intensities 2.2.1 Fluorescence and Submultiple Wavelength Diffraction 2.2.2 Parasitic Scattering 2.2.3 Compton Modified Scattering 2.2.4 Temperature -Diffuse Scattering 2.2.5 Bragg Reflections and Huang Intensity 2.2.6 Conclusions ORNI - AEC - OFFICIAL ----- '-'. - 2.3 Precautions in Crystal Preparation and Surface Contamination 4 mate .ORNI ~ AIC - OFFICIAL I 2.4 Geometry of Intensity Measurement in Reciprucal Space 2.5 Warren'e Doubly-Bent Monochromator 2.5.1 Geometry 2.5.2 Volume Element of the Monochromator in Reciprocal Space . 2.5.3 Selection and Preparation of Monochromator Crystals 2.5.4 Conclusion ORNI - AEC -- OFFICIAL QANL - AEC - OFFICIAL METHODS OF ANALYSIS FOR LOCALLY-ORDER-HD ARRANGEMENTS AND ATOMIC DISPLACEMENTS BY X-RAY DIFFRACTION* C. J. Sparks and B. Borie Metals and Ceramics Division, Oak Ridge National laboratory Oak Ridge, Tennessee 37831 INTRODUCTION X-ray diffraction is the most direct method available for determining the local arrangement of the atoms in a solid solution. The theoretical and experimental techniques for the determination of the local-order arrangements and displacements of atoms in solid solutions by X-ray diffraction have reached a state of development in which they can be applied in a rather straightforward way. Yet they have been used only a limited number of times. Most of the theory has been developed and published in the scientific literature in the past fifteen years. We will endeavor to bring out the more practical aspects of these diffraction techniques with the hope that more interested scientigts will apply them to better understand local atonic arrangements. . ". ORNL - AEC - OFFICIAL *Research sponsored by the u.8. Atomic Energy Commission under contract with the Union Carbide. Corporation. 2 There has been a l'eview by Warren anu Averbach (1) in 1953 of these ORNL AEC - OFFICIAL diffraction techniques. More recently Guinier has given a particularly readable treatment to the diffraction theory of local order and crystalline imperfections (2). Guinier has given a general introduction to the fundamentals of x-ray diffraction theory necessary to the understanding of the more specific diffraction theory which he develops for application to inperfect crystals and for this present treatment. With these reference works available, it is not the intention of the authors to repeat all the development of the theory. It is intended that later developments will be discussed and that specific treatment be given to facilitate actual experimental usage. Much emphasis will be given to laboratory procedures for the collection and manipulation of intensity data necessary for the fullest interpretation by the theory. We will follow the treatment developed mostly by Professor Warren ... . and his students at M.I.T. This approach 18 more direct and amenable to quantitative interpretation. 1. THEORY 1.1 Introduction It was pointed out by von Laue (3) that when two kinds of atoms are randomly distributed on a lattire there will be a monotonic diffuse intensity expressed in electron units per atom as . . . . . . . . . . - - XXgcea - Pipe . where the alloy consists of an atonic concentration of A and B atans of XX and Xg with atomic scattering factors f, and fp, respectively. ORNI - AEC - OFFICIAL. ORNL - AEC - OFFICIAL - - - - Not only did von Laue assume that the A and B atoms were randomly distributed, but he also assumed that the atoms are precisely at their lattice sites. With the observation of long-range order in certain alloys, * it became clear that in at least some cases an atom may have preference for certain kinds of near, neighbors. Also, because atoms are of different sizes, they may be pushed siightly off their normal atomic sites due to a noncentrosymmetric arrangement of their neighbors. These two effects cause the Taue montonic diffuse intensity to be modulated. Thus a study of how this diffuse scattering is modulated could lead to an understanding of the long- and short-range interactions between atoms of an alloy. The expressions incorporating these modulating effects on the Laue monotonic diffuse scattering will be given in this section following the treatment gaanone by Warren and his students (1, 4, 5, 6). 1.2 Order Diffuse Scattering In this section, we assume the atoms are precisely on their lattice sites and treat only the fact that they are not randomly distributed. The total coherent intensity in electron units is given by the well-known formula 15•(B, B) > ffe (1.1) eu لا لا لا لا . ma where for and f, are the atomic scattering factors for the atoms occupying the sites m and a separated by the interatomic lattice vector (-B.) and the vector k = 21 (S - )/2 with & and being unit vectors in the directions of the scattered and incident beams of wavelength , * See Barrett, C. s., structure of Metals (McGraw-H111 Book Company, Inc., 1952), Chapter XII for a good historical introduction to the Bubject of order in crystals.. ORNI - AEC - OFFICIAL . I : OAN - AIC - OFFICIAL 1 bu .. respectively. The gums are to be taken over all possible pairs of atoms that can be formed in the crystal with a separation of (A-R), 1.e., all pairs formed with each of N atoms irradiated by the incident beam. In performing the double sim, we shall always view this as a summation over pairs of atoms such that the indices n and a differ by a constant, Since the x-ray beam averages many pairs, we are able to recover only this average. Thus we should keep in raind that en and (R - R) are averages to be taken over all pairs of atoms that cau be formed in the crystal for a constant value of m -.. If we remove from the double sum of Eq. (1.2) all those pairs for which m = n, we obtain the sww over all A atoms with themselves plus all B atoms with themselves plus the term for m n. If there are N atoms in the crystal, there are NXA (A – A) pairs and NX (B – B) pairs when (Bon-Box) = 0. Therefore, Feu = 5(x,* * *99+ 2,247 * (2.23 m*n The *irst teria of Eq. (1.2) represents the average value of the total (I total, coherent intensity from our alloy, and the second term is just the phase factor which distributes this intensity in reciprocal space. We are interested part in the modulation of the diffuse intensity caused by the order among the atoms which may exist for only tens of unit cell dimensions and not the fundamental crystalline Bragg reflections. Hence we will want to subtract from Eq. (1.2) that intensity associated with the fundamental Bragg reflections. These reflections are independent of the state of order in the crystal. They depend only on the average composition and are given by ORNI - AEC - OFFICIAL *. - - Frand.-> (42A + 29 2017 **CUt! • (1.3) ORNL - AEC - OFFICIAL ma " the in If we subtract the average value (I) Fund: = N(XAPA + X3 B)2 from the average total intensity (I)total = N(XALÁ + x B). We have (I) pipruse = N(XAFA + x3): – N(XAFA + x B)2 = NAX342A - fig)e . (1.4) This is just the Laue monotonic diffuse scattering for a random distribution of A and B atoms. This intensity arises from the fact that there are two kinds of atoms present and depends on the difference in their scattering factors. We will now introduce the concept of order into Eq. (1.2) ving the conditional probability PAD = the probability of finding an A atom at the lattice site m after having found a B atom at n. After rearranging a few terms, we shall subtract the intensity due to the fundamental Bragg reflecticas leaving the Taue monotonic diffuse intensity modulated due to the ordered arrangement of the atoms. The term XFAS is the probability of reaching into the crystal at randon and finding a B atom at site n then finding an A atom at the site m. Since the number of (B – A) pairs must equal the (2.- B) pairs, xem = x pena. Also, PER - 1 - PR and Print - 1 - Bro. We want to evaluate the second term of Eg. (1.2) using these conditional probabilities. We assume N large enough to neglect edge effects. To include all possible pairs, for 1s evaluated for (A - A) pairs, (A - B) pairs, (B - A) pairs, and (B - B) pairs separated a distance (Bx - B) such that boniere om o mamme .. (A) - se mere gent gran. (1.5) ORNL - AEC - OFFICIAL ..:- ------ * ORNL - AEC - OFFICIAL Using this relation for info in the double sum of Eq. (1.2) and substituting PB = A * MATA = 1 - MA and PE = 1 - / we have that Ieu = N(XAFÂ + xy fee) (1:6) + [Z[<(: - ) ** + expert op * x367 - para 123] ****a! min At this point we wish to separate from Eg. (1.6) the term representing the sharp fundamental Brage reflections that are unaffected by order. We subtract Eq. (1.3) from Eq. (1.6) to arrive at the intensity Iorder due to order in the alloy, Teu - Spune. - Forder = x, y en – sve + 22(4,62 - + expert op **(- - (4,90*40*] ****! (1.7). By rearranging terms and with the substitutions Xx - XX = XAXBO and X3 - X = X X3, Eq. (1.7) can be written as Forder = NXA X BIPA - PB)2 +2 2**(A - fb)2(1 – P/XB)e * min B . (1.8) maa Finally, the order coefficient is defined as Con = 1 - PA/Xgeand the first term of Eq. (1.8) 18 placed under the double summation giving Forder -2 E4%(CA - file *.!? (1.9) m a OPNL - AEC - OFFICIAL · OANI - AEC - OFFICIAL : This result was first given by Cowley (5). It is completely general and may be specialized for a binary alloy of any crystal structure. The oruly assumption is that the atoms are precisely on their lattice sites. The static displacements of atoms from their sites will be treated in a later section. Since min = 1 - PA/Xg, for a random distribution of atoms un = 0, but doo = 1 in all cases because peo must be zero as one cannot find a B atom on the same site as an A atom. Walker and Keating (7, 8), however, have pointed out that thermal vibrations of the atoms also effect the order parameters. The man's are reduced by an e 2M® factor where 2M 18 the usual Debye-Waller factor and ™ a function of (Rm - Rn) being slightly different for each coordination shell. The lower-order Own's are less affected than those of higher-order, and 1f the intensity measurements are made in the region of the first reciprocal unit cell at low temperatures the effect is small. For the case of CugAu with intensity measurements made at 405°C near the (100) superstructure reflection, Moss (9) found this temperature factor affected the ain's by approximately one percent. When there is a preference for the atoms to choose unlike nearest neighbors such as A likes B, this is referred to as short-range order. In this case the diffuse intensity is distributed away from the fundamental Bragg reflections. If there is a preference for like nearest neighbors, it is termed clustering and the diffuse intensity is found nearer the fundamental Bragg peaks hecause the first nearest neighbor a will be positive. For alloys having a strong tendency to ciuster, low-angle scattering techniques would be necessary to measure the intensity distribution, . . ORNI - AEC - OFFICIAL .' 4 - " . '. -'in -, L SA . . . These order coefficients can also be used to describe long-range order as shown by Cowley (5, 10). In this case their absolute value would be a maximum. They are related to the Bragg-Williams (11) order parameter S which is proportional to the square root of the intensity of the superstructure reflections. The long-range order parameter S 18.one for the fully ordered case and zero 1f the atoms are distributed at random.. For complete long-range order, all the Laue monotonic diffuse ORNL - AIC - OFFICIAL intensity is gathered up into the sharp superstructure reflections. When there is only partial long-range order, that part of the Laue monotonic diffuse scattering associated with the superstructure reflections 18 52. Thus 8 is a measure of the wrongly occupied sites, Chipman (12) separated the superstructure reflections from the diffuse background for a partially ordered Cuzau crystal and was able to determine how the wrongly occupied sites were distributed. In the next chapter Moss and - r . Schwartz give a more complete discussion of partial long-range order.. 1.2.1 Special Case of Cubic Solid Solutions Because most measurements of order diffuse scattering have been made on cubic bine y solid solutions, we wish to examine Eq. (1.9) for that case the particular. To express the diffuse intensity for a position in reciprocal space in terms of the atom positions in real space, we represent the vector from one atom to another by. · so that l, m, n are integers for the face-centered cubic Aj and body- centered cubic Ag type structures and 21, 22, ag are the usual cubic axes. . * .* .. .. 46 ORNL - AEC - OFFICIAL . 7 . Reciprocal to ten, kurz, and the things are the vectors bon, bz, and bs which are twice the magnitude of the usual reciprocal vectors. The diffraction vector times 21 1s then represented by OINE - AEC - OFFICIAL ($ - $) $ 2 21(1761 + habe + babes) where ha, ha, and bg are continuous variables in reciprocal space, Figure 1 shows schematically the atom positions in real space and the associated reciprocal lattice cell for the face-centered cubic lattice. The exponent in Eq. (1.9) can now be written as -- ---- - w w lk. (B - .) = 2171(hıl + ham + hgn) . Since for local order the coefficients Own converge rapidly for increasing values of men in an interval small compared to N, the double sum of Eq. (1.9) may be replaced by N times a single sum taken over neighboring atoms of coordinates l, m, 2. The total number of atoms in each coordination shell, C, 18 given by all the combinations and per- mutations of the indices th, tm, in; therefore, the sums over l, m, n extend from co to do. Hence, we write Eq. (1.9) as ro 2011 (hyd + ham + bgn) Iorder = NXA (A - 3)2 Llamne . (1.10) mantel-retro-mediante As a final simplification of this equation, we know that for each aton at l, m, n there is an atom at the coordinates T, m, a which has the same value of a . Thus the sine terms cancel and Eq. (1.10) can be -. '. , -' expressed as - - Ipo - NX 33/PA)2 2 2 Camcos27(hyb + ham + hgh).' (1.21) order This expression 18 Just a three-dimensional Fourier's series with the periodicity of the reciprocal-lattice unit cell. -ORNI - AEC - OFFICIAL --- P3 10 criminaitwa ORNL ALC - OFFICIAL .. ..... . . 1.3 Atomic-Displacement Diffuse Scattering The assumption was made in the derivation of the order scattering · equation that the atoms are precisely on their average lattice sites.. It had been suggested by von Laue (3) and Guinier (13) that the atoms may be slightly off this average lattice due to their difference in size. Warren, Averbach, and Roberts (6) have treated this atomic size : : effect by allowing the interatomic vector for a given coordination shell to be slightly different for the three kinds of atom pairs. The treatment pestenent here 18 a generalization of their derivation as qiren by the authors:614). Previously (Rm - R) was defined as the vector difference between atams. By measuring the 20 positions of the Brage reflection and knowing the crystal structure, one actually measures the average interatomic distance (m in). (Em - ) = (A.- B) 1f there are no static displacements of the atoms from the average lattice. But 11 the atoms are pushed off their average lattice positions, an additional term is added such that. ( R B ) = (en + Oman (1.12) where &mn represents a small vector (om -&n). Figure 2 shows this relationship schematically. The average lattice is drawn with (Emrin) being the average interatozaic vector. The small vector mm accounts for the displacement of the atoms off this average lattice. The atomic displacements are accounted for by replacing (R.- B) 10 the exponent of Eg. (1.2) by (- In) + Amin giving ......... tot moment -- men . . : Feu - (3, * *43872 +247**i) + Remune]. (1.28) 1 Le (1.13) ORNI - AEC - OFFICIAL - PR IZ Thus for each pair probability Pon which is the probability of finding a B atom at m from an A atom at n, the average separation distance (R.- B) for thise (A - B) pairs may not be constant and is replaced by (open - Eine + or A is the average value of n ex) for all (A - B) pairs separated by a constant (Im - Ind. Using the same arguments as previously to arrive at Eq. (1.6), Eq. (1.13) can be written as · ORNI - AIC - OSSICIAL Lou M4*** +22x61 - O (fea) * @(.24) min .-.-.- - - -- With the assumption that lk.mn 18 small compared to one, the exponential term may be approximated by _ . kelin ton) * Run 2 o 4. in fine (2 + tesland a n .. - - and on substitution Eq. (1.14) becomes Teu = (x* * * * (1.15) +2%E163 - * * * * * *(1-x2)5]. ***! +2[[1,6 - 561 K.8° menormomimme *Lifik.89 a A B ann X 3. mana The first two terms of Eq. (1.15) are the same as Eq. (1.6) and the third term 1s the atomic-displacement sum. On subtracting the intensity ORNI - AEC -0$$ICIAL ! Pod . Yah. - . associated with the Bragg reflection as given by Eq. (1.3) from the first .. -. . ORNL - AIC - OFFICIAL ..: . two terms of Eq. (1.15), we get the order term identical to Eq. (1.12) plus the atomic-displacement term as follows: kou - Hrund. - Forder * Fav - E24,684 - Balaton * tot (1.6) + 2%[1,c2-pdf.pushed to exploit moment :(2 - gutega) *** B. There is a relationship between rechte in and out for any coordination Pielik 8 Amn AB No min : shell mn taken abbut every atom as the center. We must on the average maintain the average interatonic vector (Im - In) in order to conserve volume. Therefore, the weighted average of the distortion on for all (A - A), (A - }); (B - A) and (B - B) pairs 18 zero for any coordination shell. Hence we write AAAA BAB X DAB : mom Bmmmn Rewriting in terms of An tomado BA en BA AB . , we have that 2x -- [x (2 pay + (1 - R 29 m.). (2.17) : . . Substituting this value for 2x porno in the second sum of Eq. (1,16) and rearranging terms, we can write for the atomic-displacement term only ma 19-2X (A - Pg XX9 ** * * * momentet (15 - * * ay ) (1.18) *** ORNI - AEC - OFFICIAL in on multiplying Eq. (1.18) by XA (FA23)/*A*3(A - Pg) and extending the double sum to include the term for n - n which 18 zero since an atom cannot be displaced from 1tsell, we have im- Exc.- solo contrat (1.1) (A - fB? : No Intensity has been added or subtracted; the Laue monotonic is only restrictions on how the displacements of the atoms off their average lattice occur except that they be small. Of course we can measure only the average value of the displacement on and this does not say for example that all (A - A) pairs separated a distance (B. - Bx) have the seme SMA Warren, Averbach, and Roberts (6) point out that the atomic-displacement term may not be strictly valid in the case of appreciable segregation, One could calculate arbitrarily large values for the displacements depending on the definition of the average interatomic distance TEM – Enl. In the extreme case of complete segregation the A atoms would have an average interatomic distance of the A lattice only and the B atoms of the B lattice only. This could lead to a broadening or splitting of the Bragg peaks. To use a weighted average of (in En) AA and (r. - Im BB would imply a fictitious strain. The approximation has been made that the exponential term * mm m 1 + 1k•8mHowever, if the next term in the series expansion ORNI - AEC - OFFICIAL 14. i . OINI - AIC - OFFICIAL . . n partenar 18 included, it causes a thermal motion-like effect on the order peaks reducing their intensity. This reduces the values of the Qur's by an (-2M + Enn) W! factor similar to the reduction associated with the Debye- Waller factor e M. This factor has been derived theoretically by Borie and is associated with root-mean square static distortions. The effect on the annis is small and most likely beyond experimental detection within the first and second Brillouin zones of reciprocal space. There are other poseíble errors in the measurements as will be pointed out. later, which make for larger uncertainties than neglecting the effect of (-2M' + L ) man' on the ann's. If we make the assumption of Warren, Averbach, and Roberts that Omn is on the average parallel to the interatomic vector (. In) then Son can be replaced by (open - Lindeman where Emn 18 a small scaler parallel to (Ene - In)On making this substitution in Eq. (1.22), we write the sum originally termed the "size-effect" sum by Warren, Averbach, and Roberts as - - . omination concentración con - .. i atsiooniminister in : n memories ----- ISE * 2XAX (A - P3) Banko (- Fine (1.20) momento non --... -- ma - - where the size-effect parameter 18 defined . Panel de Sunder. (1.21) - - ume mama - - - . marami na - . - - BBS Jemn B • (1.21) 06) This result was first given by Warren, Averbach, and Roberts (6). Guinier (10) has suggested that the displacement of the atoms may not necessarily occur on the average only parallel to the interatomic vector. His prediction has been experimentally verified by the authors (15). Since the Lininiai : displacements of the atoms off the average lattice need not be strictly vidijo - 33V1N10- ... . 7.1'ini... the result of the physical size difference between the atoms, the authors use the term atomic displacement to denote this effect. on replacing the relationship given by Eg. (1.21) by Born in Eq. (1.20) we are assuming Ben 18 independent of the vector k or the distance from the origin in reciprocal space. For this to be so, the ratio of I/12. - fe) and fp/(A - 13) should be constant. Though the atomic scattering factor decreases with increasing sino/^, this ratio is nearly constant for short intervals. 3.1 Special Case of Cubic Solid Solutions Just as in the case for order, we will write the atomic-displacement equaticü in terms of the atom positions l, m, n and the reciprocal lattice Indices baz, ha, hg. For the face-centered and body-centered cubic alloys, Ik• ( m) = 2011 (112 + hom + hen). Omn may also be written in terms AA 7 A I . of its components along the cube axe to mind Zvan 23 and of course + + The factor į is used since we specify that l, m, n are always integers. Therefore ikone = 2011 (12 + hamin + horm) and similarly for ik. Pl. On making these substitutions in Eq. (1.19) and replacing the double sum by N times the single sum over l, m, n, we write Fav * *,500 - $) ? 2. Prace - wyda ASO + 4 punkti (1.2) 2 mn A . i(hyl+hanth, e le termen de casa conveyzul)fazakartan) hone mn im Belmo 111911:0-31 -156 16 7:13!!:)3 For every atom at eman there is another atom at I mnā. Since ik edm = -ikoma, then Lemn - - Inn, Memn - Momā , and Nemn = -Non, the cosine terms cancel and Eq. (1.22) can be writtea as follows: 1 :1:) IAD = - NXX3(PA - PB)2 2 2 2 (127émn + barem + hormo) sin21 (12 l+hamthega) (1.23) & m n and I am the problem thon - a to na beton (1.24) with similar expressions . Here again the assumption is lmn . ., . made that IA/(afp) and fg/(PAPs) are constant. IP Elma is on the average parallel to the interatanic vector im Einde then Eemn = egun beren - Em) = (egmmlžan + € granmgaz + Egmongas) and L'Armin = løgmn? Mema = me emn' and Nema = neem. Thus Penn E l Lo + aligne) er et + apminderen ), and from the definition of Bemn given by Eg. (1.21), we have that b27emn = 2uthy eBeman har man = 212mb emne and hasrema = 2th gap emn. Therefore, when the displacements are on the average parallel to the interatomic vector Womende = vo mn/m = nomin/n and reduces to the Warren-Averbach-Roberts size-effect sum . . . . . Ex. (1123) ooo ... · Ise = NXAXB(PA- ...). 6. 201(196+ ham+ hän) sin2(hıl + ham+ hgn). Land emn (1.25) 7:1011) - ..1:::) . -:::... 7.111:: Since on the average the alloy is still cubic, lemnl is the same for all permutations and combinations of £l, tm, in. For the highly symmetrical directions d = m, n = 0; l, m = B = 0, and f = m = n, Sem on the average must be parallel to the interatomic vector to maintain statistically the cubic symmetry. If no has a component M20 > 1910, the average wit cell dimension sa would be larger than az as shown in Fig. 3 in two dimensions. However, for l * m, n = 0; d = mka, and l + m n, Elin need not be parallel to the interatonic vector. For these three kinds of directions the multiplicity is 24 or greater, and there are always at least two different pairs of directions symmetrically disposed about each cube axis. Therefore, the vector sum of the components S along each cube axis is the same even though the displacement may not be parallel to the interatomic vector. This is illustrated for 8210 in Fig. 3 where a rectangle enclosing all <310> displacements is square even though the displacements are not parallel to the average interatomic vector. Taus the assumption that the displacements are on the average parallel to the interatomic vector must hold for some directions but is not necessarily so for directions of multiplicity equal to or greater than 24. On recovering the coefficients nomen as outlined next, we can determine if the displacements are on the average parallel to the interatomic vector. Recovery of Order and Atomic-Displacement Coefficients of the Diffuse Intensity 1.4.1 Separation of Order and Atomic-Displacement Modulations Because the measured intensity data includes both the order and atomic- displacement series, we must separate them to recover the parameters @gma and Yemno For a fully long-range ordered alloy, the atoms will be precisely il-iis-nisiri 18 7:10;;:). on their lattice sites. Therefore, atomic displacenient modulations exist only for locally ordered or rendon solid solutions. The laue monotonic diffuse intensity modulated by the order and atomic-displacement series - 1. 0 specialized for the face-centered and body-centered cubic solid solutions is the sum of Eqs. (1.14) and (1.26). This is given by Forder + AD = : NX33(1Afg)e ...comincos21(57.+ ham + hen) (1..28) [hayemn + har man + her om Isin21 (12 +ham + hgn) . ļ ma Borie (16) has pointed out that because of the different symmetry of these two series they may be separated. We extend his treatment to include the three-dimensional case. Define the function Q(b1,b2, 23) by I(til, ha, ha) - I(hq, ha - 1, he) Q(h1, h2, h3) = - NXXB(A - FB) (1.29) Since l, m, n are integers, cos27r(bąd + h2mm + hgn) = cos2r(€ + hgw+bori) and the local-order series is subtracted out leaving Q(hi, h2, h3) = > alba, ba,he) -- 2 2 2 (napámn + bareme * bayam de 1 metr(hyl + bun + han) (.30) 117 emma + 2y ema. 37 ima 2T (Hand + hem of hen) Waw +22 Einayan batin - Pimn + har man detrær (jag babaganathem). Because sinzir(h1£ + ham-m + hen) = sin (112 + ban + hgn), Eq. (1.30) can be written as Tii !:) .. ...... .:) 19 Q(hı, ba, bg) = om sin2m1 (bąd + ham + bgn). (1.31) ??!01!:0.337.-1:::::) Notice that Eq. (1.31) is just a three-dimensional Fourier series with the atomic-displacemenü parameters zemn as the Fourier coefficients. Thus ve have our data in a form such that the atomic-displacement coefficients can be recovered. With reference to Fig. 4 where the normal to the crystal surface is represented by the arrow marked N, we chose to make the separation along ha so that the variation in the ratios PA (PA IB) and 23/(PA-23) is minimized. If the separation of the local-order from the atomico displacerent series is made along hi, the difference in scattering angle would be a maximum. The function Q(ha, ha, hg) is formed by choosing any point Ilha, ha, hz) which we call A in Fig. 4a, then going minus one translation distance in ba to point B. There must be symmetry for the total diffuse intensity distribution across all planes of the type 100 and 120 which cut the origin of reciprocal space if the alloy is still statistically cubic. This is the only restriction on what symmetry must be present. Since there is symmetry across the plane (hi, O, hg) for the total diffuse intensity, the measured I(bi, ha-1, bg) at B is the same at B . Also the intensity at B' is the same at B" because the intensity is symmetrical across the plane hı = ha, hg. On subtracting the intensity at " from the intensity at A We have determined a value of the function Q(h1, h2, hg) for the position A in reciprocal space. If this procedure is repeated for every datum point in tihe cube, the resulting function Qlba, h2, h3) has at least the symmetry shown in Fig. 40. O'::!! -;.??. - 0:1!C!:.L 20 To recover the Incal-order series only, we need to reconstruct the 1713131) - 338 - - - --. atomic-displacement series from the Q functions. When Q(11, ba, h3) is multiplied by ha, there results. . - - , 1:1:10 . ico healba,ba, hog) - 22 na p ostment (1946 + baw + bgn) (.32) which is just one of the terms for the atomic-displacement sum as given ---- . ... in Eq.(1.28) Atiother term containliig yam can be obtained by subtracting the intensity at Ilh1, h2, h3-1), point c in Fig. 4a, from the Intensity at I(hı,b2,13). With reference to Fig. 4a, we see that this procedure is the same as interchanging the indices ha, hg because of symmetry across the plane (hi, ba = 13). This results in the term - Saum e CO - •- -.---.. . . . (ha, ba,ba) = -2 2 2 Ammeiner (h32 + bam + bon). (1.33) .. ... . Similarly, the function Q(ha, bi, ha) can be made by the interchange of ' the indices by and ha such that Q(ha, h2, h3).=-).). ).mmrsin2 (17! + ham + hen). (1.34) mn & Ian on multiplying the functions Q(21, ha, ha) by hg and Q(ha, bi, ko) by ha and surring all three, we have AD .. NX .- (PAFB) = ya = hiQ(ha, bı, 13) + haQl11, h2, h3) + hgQ(hi, lg, bal (1.35) .-. -.- - - ... (hayamon t har man + hogy mm) sin2(b14 + ham,+ bga). į ma 7:13!1!0--1: 21 77131:3u - 138 - 1:15:0 Thus the original atornic-displacement term as given by Eg. (1.28) has been reconstructed so that we can subtract the intensity represented by Eq. (1.35) from the intensity distribution represented by Eq. (1.28) and are left with the order series only. This separation of the order and atomic-displacement series can be done in a volume of reciprocal space enclosed by the heavy lines as shown in Figs. 5a and 5c for the fcc and bcc structures. In Fig. 5b, the additional symmetry in the Q function over that drawn in Fig. 4 is because yon = me and, therefore, the function Qlh1, h2, h3) nas symmetry across the plane (h1 = hg, ha) for the face-centered cubic case. For the body-centered cubic case shown in Fig. 5d, the fact that l, m, n are either all even or all oda leads to even higher symmetry for the function Q. The order series derined by Eq. ( or cubic solid solutions can be written as more pre ss - 2 2 2 Oma cosewall costban coserbga (.36) & m n mm mn Pernamn because of the fact that a lmn = Qarn The Fourier transform of Eq. (1.36) is obtained by multiplying both sides by the cosine terms and integrating over the repeat unit in reciprocal space. On replacing the integral by the sum, we can write I i I > I(h1, h2, bs) (1.37) hq=0 ba=0 bg=0 ayn [164509]* 2 Xcos2thyl cos2ham cos2th 32 Any shadhg . 0:.::1 -:1.0-057:01:51 22 . by ChiL::C - Criclit ..... - - - - . . - . ** I(h1, ha, ha) is appropriately weighted by 1/8 if it is at the corner of the cube as it is shared equally by 8 cubes and similarly 1/4 por edges and 1/2 for faces. Because of the synometry for the diffuse intensity associated with the ordered arrangement of the atoms, the series need not be evaluated over the whole repeat unit of hq - 0 to I, ha = 0 to 1, and bg - 0 to 1. If the crystal is statistically cubic, Clemn will be the same for a particular coordination shell for all combinations and permutations of Il, tn, and In. Figure 6 shows the minimum repeat wat in reciprocal space for the order diffuse intensity from a face-centered and body- centered cubic solid solution. Thus, we need only to perform the summation of Eq. (1.37) over this small repeat unit which is 1/96 of the reciprocal unit cell for the face-centered cubic lattice and 1/192 for the body- centered cubic. If the diffuse intensity is measured in a repeat volume in reciprocal space, the order coefficients Comn can be recovered. Of course, these a's are averaged about all A ir B atoms as the origin and do not imply that all A or all B atoms have exactly the same environment as this would mean long-range order. The values of the indices of the atom position 2,m, n are such that ļ + m + n = even number for the face-centered cubic lattice and are either all even or all odd for the body-centered cubic lattice. The twelve first-nearest neighbors in the face-centered cubic case are designated 110. On recovering (110 from a Fourier inversion of the intensity data, we know how many of these nearest neighbors to ar. A atom are on the average B atoms. .::!~!!!C- O:TICI.L 23 To recover the atomic displacements coefficients a w We write ORAL - AEC - OFFICIAL Eq. (1.29) as Qaz, h2, h cosmhid sinham costruge. emn This result can be easily verified by using the fact that por men women Yemn - - = Pemen - in = - ponto = pi ma because the alloy is statistically cubic. The symmetry expressed in Eq. (1.38) verifles the symmetry for the function Q(h1, h2, ha) drawn in Fig. 4b since there is cosine symmetry along hi and hg but sine symmetry along ha. We can do the Fourier transform of the function Q(21, ha, hz) in a fashion similar to that described for the order series. With the point hi, h2, ba - 100 taken as the origin, the Fourier transform of Eq. (1.38) is 1 Yemn 2. Qlha, h2, h3) cos2rhad sinham cos2hgn 02. Cazane (1.39) | ht=0 ha=0 ht=O where the sum is taken only over 1/8 the large repeat unit in reciprocal space. wmn and women are obtained by interchanging & and m or m and n, respectively, but no additional information is gained. In writing Eq. (1.39), we have assumed that each corrected value of the measured Intensity at hı, ha, kg has been converted to absolute units and divided by NX,X (fa-fr)? for that position in reciprocal space. In Sect. 2 we will see how these corrections are made, On recovering the order coefficients @gmy and the atomic-displacement coefficients homme we have obtained the maximum amount of information available about the system without having made any assumptions about a specific model for the arrangements of the atoms. It is much easier to ORNL-LEC - OFFICIAL 24 .. fit a specific model of the arrangements of the atoms to these parameters . - . . . than to try and match a model to the diffuse intensity distribution directly. If the two distortion parameters Man and MBB contained in the recovered coefficient namn could be separated, even more information would be gained about the alloy without recourse to a specific model. . . . . . . .. . .. . In the next chapter, Moss and Schwartz will cover recent efforts to fit specific models of the local-order arrangement of the atoms to the comin's. From a measurement of the a's we have a complete description of the number of (A-A), (A-B), and (B-B) pairs for all l,m,n coordination shells. But the atomic displacement parameters namn contain two terms ..... .--. BB which are the displacements of (A-A) pairs and and Me einn lirs enn . along the cube axis. It is not known at present how to recover these two terms separately. - To fit a specific model to these atonic displacements parameters is consequently more complicated than fitting the a's. To date no three-dimensional measurements of the atomic- displacement parameters have been made, but two-dimensional data have been discussed by the authors (45). Any acceptable attempts to fit a model to the atomic-displacement parameters will surely require three- dimensional data. It could be that the additional information gained from the atomic-displacement parameters might determine the plausibility - ----- . - - . - - - - - - - - of certain specific models based on the a parameters only. 14:3 Treatment of One- and Two-Dimensional Data - - - - - It has been rather common practice to only make measurements of the diffuse intensity along lines or in planes of reciprocal space because of the time consuming process of data collection for three- dimensional data. The three-dimensional equations can be easily specialized for one- and two-dimensional data. 00:!! -- 16 - 03F101. ORNI - FC - OFFICIAL . i 25 For example, if we make intensity measurements in the ha, ha, 0 plane of reciprocal space, the local-order and atomic-displacement modulations of the diffuse intensity as given in Eq. (1.26) can be written as ORHL-AEC - OFFICIAL I(h,,ha, 0) in 2. Agm cos211 (h] [ + həm) - (hytém + her terme) (1.40) NX X3 (FAB! ē mm & m x 51127(hoz+ham) where Aer = 20 cm Dae 7 The local-order emns and Me emn • n and atomic-displacements terms can be separated as before if the intensity distribution is measured in the area outlined by ABCFD as shown in Fig. 5a. The two-dimensional parameters Apm and them which are combinations of the three-dimensional parameters can be recovered by a Fourier transform of the corresponding two-dimensional representations of Eqs.(1.36) and (1.38). As an example of the combination of the Cama's represented by the A 's in the hy, h2, 0 plane for the face-centered cubic lattice, we have em that . . . . - . - .-.- - -. . Ano = 4000 + 20200 + 20400 + 20600 .... A10 = 20.110 + 28310 + 20510 .... Az1 = Cyno + 20211 + 20411. + 20611 .... A20 = 0200 + 20220 + 20420 + 20620 .... - ..-. - - .- -.. - .-.- -- . -.- . .- - - -- . A60 = 0600 + 20620 .... ORNL - AEC - OFFICIAL priekinian wanita berkenaar wins r 26 until A. converges to a value no longer considered significant. It is ORHI - AEC - OFFICIAL very questionable practice, however, to evaluate the comin's from the above combination. There are always more unknowns agmn's than known Aom's. Even if measurements are made in two planes in reciprocal space *Lma with the number of known Agm's almost doubled, one cannot uniquely determine the three-dimensional parameters. By assuming a value of [ + m + n above which demn = 0, the unknowns can be reduced sufficiently so that the relations can be solved. It has been the experience of the authors (14) that the three-dimensional a's recovered from two-dimensional data can vary markedly depending on which a's are assumed small, the. weighting of the Agm's depending on their magnitude and accuracy, and the assumption that for the measured data d..= 1. Therefore, it would appear that the best approach is to fit the specific model of the local-order arrangement of the atoms directly to the two-dimensional A's until more accurate measurements are made in three dimensions and atomic displacements effects separated. Until an evaluation of errors is made for that case, it is difficult to estimate the errors inherent in two- dimensional data. A worse situation exists of course for one-dimensional data since the recovered one-dimensional A, 's are combinations of larger numbers of Cen's. Walker and Keating (18) made measurements along more than one direction in reciprocal space and concluded that the comp's could not be recovered with sufficient accuracy from this one-dimensional data. Schwartz and Cohen (19) made intensity measurements along the 100 and 210 directions in reciprocal space for a CuzAu crystal with short-range order. They expanded the measured one-dimensional coefficients in terms of the HL-AEC - OFFICIAL 27 three-dimensional ads and determined sais associated with annealing at 100°C. There is great difficulty in trying to estimate the error involved ORHL-AEC - OFFICIAL iere. But for a rapid evaluation of relative changes in the local order, this approach could have an advantage. Of course one must choose those 1 . . - . .-. --.. * . .- -- - directions in reciprocal space which contain the diffuse intensity maximum for most information. 1.5 Application of Theory to Powder Samples Because of the difficulty of growing single crystals of some alloys and for expediency, measurements of diffuse intensity from powder samples are often made. In the most favorable cases where there is a strong tendency for local-order and a large ſaue monotonic diffuse intensity, one might verify this tendency for local-order in the alloy as was done for example by Houska and Averbach (20) for Cu + 14.5 at. % AL. For a powder with no preferred orientation, the intensity distribution in reciprocal space is averaged over all possible orientations about its origin, which means that the interatomic vector (RR) has equal probability for all orientations relative to the diffraction vector k/211. We quote the result of this average given by Warren, Averbach, and Roberts (6): . .. m i 7.-.-. Ani . . . ... . - - . --.. --- ........-- Nx3(PAPse where C, is the number of atoms in the 1th coordination shell, ra is the radial distance of the 1th shell and S = (4 sine)/n. Rather complete discussions of the use of this equation for the interpretation of the diffuse intensity from powder samples have been given by Warren and Averbach (1), Flinn, Averbach, and Rudman (21), and Kaplow and Dupouy (22) ..wam-- world where time. c RNL - AEC - OFFICIAL e h 28 and need not be repeated here. The detailed structure of the diffuse OSX1 - AEC - OFFICIAL intensity distribution is lost because of the averaging of the intensity over a spherical shell about the origin and a precise interpretation in terms of the order and atomic-displacement parameters is not posssible. to Intensity Expression for Huang Diffuse Scattering In the derivation of the atomic-displacement diffuse scattering . equation, the exponential term ek man which accounts for the atoms not being precisely on their lattice sites was approximated by the first two terms of the series expansion. If the first three terms are included such that "^1 + ikem - 1/2(komme, (1.42) the term 1/2(komna gives rise to a diffuse intensity distributed about the Bragg reflections. That such a diffraction effect should exist (23) for alloys with atomic displacements was pointed out by Huang (22). The quadratic term Eq. (1.42) is a measure of the spatial fluctuations in interatomic distances and is similar to the fluctuations in time associated with thermal motion. Huang's theory showed the consequence of this variation in interatomic distance to be a reduction of the integrated intensity of the Bra£g reflections by a factor e 2M similar to the. Debye-Waller temperature factor but associated with the static distortions. Borie (23, 24) was able to derive this result in a more general way such that both the predictions of Huang and Warren, Averbach, and Roberts (6) were found in a unified approach. We do not intend to rederive their equations here but only to quote the results. ORNI - AEC - OFFICIAL Two important assumptions were made in order to derive the results: ORNI-ALC - OFFICIAL there is no local order among the two kinds of atoms; and the crystal lattice - - - - - is strained by spherically symmetric distortion centers acting in a continuous perfectly elastic, isotropic medium. It is further assumed that the atom deyiates from the average or undistorted lattice by the vector sum of the individual displacemento caused by each distortion: center in the crystal. Thus each distortion center gives rise to a displacement which decreases as the square of the distance from that . . - '.- center. The distortion center is either an A or B atom with a distortion ' . - .i ** 1.. ' ... coefficient CA or Cg, respectively, where C = rear. (1.43) The average radius r of the atom in solid solution is known from the crystal structure and the unit cell size. Associated with each A cr B atom is a Ar, or are which tells how much the radius of the atom in . ". ..... .. solid solution differs from the average radius r. Ar and consequently C will be either positive or negative depending on which atom is the larger or smaller in solid solution. Because reference is made to the average lattice, volume must be conserved, and x ea + X CR = 0.. Specialized for face-centered and body-centered cubic solid solutions, the total diffuse intensity In associated with the three terms of the series expansion as given by Borie (24) is minta - [1473) – (3,84 * Refplega 2 2 8CA T T 5211 (by l+ham+hga) (1.44) * sim27 (h2+ ham+han) where a, is the unit cell size of the cubic solid solution. This equation is obtained by subtracting the fundamental Bragg intensity from the total intensity. The sum over l, m, and n is large near the podes in reciprocal space being positive on the high-angle side and ORNL - AEC - OFFICIAL SU ' IC .! '. . negative on the low-angle side. Thus this diffuse intensity is ORN-AC-OFFICIAL concentrated near the Bragg reflections and increases for Bragg reflections further from the origin in reciprocal space. If the bigger atom also has the larger scattering factor, this diffuse intensity will be stronger on the low-angle side of the reflection. Further away from the Brage reflection the intensity drops to the value of the Ixue monotonic scattering. The Huang diffuse scattering is accompanied by a 21 decrease in intensity of the Bragg reflections. The intensity loss is proportional to 1-e2N in the same manner that the thermal diffuse intensity is proportional to 1 - M. Measurements by Borie (15) iind by Herbstein, Borie, and Averbach (25) show that in those particular cases the root-mean square displacement of the atoms off their average lattice site is about 1/2 to 1/3 that due to thermal motion at room temperature. From Eg. (1..44), we see that there could still he a. Huang diffuse intensity even though the atomic scattering factors to and for differed very little. Thus, with an alloy of atoms with nearly identical atomic scattering factors where no Laue monotonic intensity exists and hence no atomic-displacement modulations, one could still measure the distortion coefficient, Other treatments of the Huang scattering have been given by Srirnoy et al. (26, 27) and by Schwartz and Cohen (28) to include the case where partial long-range order is present. A result of this treatment is that there is also a Huang intensity associated with the superstructure reflections in the case of partial long-range order. OR".1.06 - OFFICIAL 31 The Huang intensity distribution is difficult to measure because of the large temperature diffuse scattering near the Bragg reflections. have To date, tbere has been insufficient experimental results to check ORNI - AIC - OFFICIAL - - - - - - - - ... - - the theory which is based on a model, though plausible, hardly expected - - - - to apply in every case. It appears that each diffuse intensity experiment should be treated as a separate case until additional knowledge is obtained on the generality of the form of the distribution of this diffuse intensity. We need, however, to be aware of the existence of such scattering so that we may better treat our measured diffuse intensity for this effect. * EXPERIMENTAL TECHNIQUES 2.1 Converting Intensity Measurements to Absolute Units 2.1.1 Intensity Expression in Absolute Units To facilitate the comparison of the measured diffuse intensity to . that predicted by theory, it is necessary to convert the measured t u isteneities I to absolute units. We define the absolute intensity Im.. eu tum in electron units as -- (1.45) where I ls the Thomson scattering from one electron. Thus I, is defined as the intensity scattered from the sample divided by the intensity scattered from a classical electron placed at the same point as the sample. I is given as, Te * Io P. (1.46) m2c4R2 In this equation, R is the distance from the sample to the detector, e is the charge and n the mass of the electron, c is the velocity of light in free space, and p is the polarization factor for the experiment. ORNL - AEC - OFFICIAL 32 . ..... . - ORNI - AEC - OFFICIAL ........ - .. - - .... Assuming the incident beam I, directly from the anode of the X-ray tube to be wwpolarized, the polarization factor p is (į + cos220), where 20 is the scattering angle. Since normally our experimental arrangement will include a monochromator of scattering angle come the polarization 1 + cos220 cos220m factor p is - m. This assumes that both the sample and 1 + cos220m monochromator scatter as ideally imperfect crystals. For a more complete discussion of polarization factors see Klug and Alexander (29). It is convenient to rewrite Eq. (1.45) as I = N (1.47) where N is the total number of atoms in the sample irradiated by the incident x-ray beam and Iy/N is the intensity in electron units per atom. However, all the atoms are not exposed to the same incident intensity I, because of absorption. With reference to Fig. 7 Where the angle of incidence equals the angle of reflection, the correction for absorption in the sample may be calculated. Iet the area (1° IG be Ani then the small element of volume containing an atoms is AV = sine dx . (1.48) If N. is the number of atoms per unit volume in the sample, then AN = N A dx/sine. The effective scattering from the AN atoms located a distance x beneath the surface is reduced by absorption through a path length of x/sine on entering and x/sine on emerging. Thus the intensity scattered by an is reduced by e Brue where u is the linear absorption coefficient of the sample. Hence, the contribution to the intensity of the volume element an is 49 ORNL - AEC - OFFICIAL _ 33 2ux av = 2 (4*) ante Moto (*14, e mais (.49) The intensity is obtained by integrating Eq. (1.49) over the thickness of the sample. Assume the sample is thick enough to completely absorb the beam. After substituting P = I A. for the power of the beam incident on the sample and integrating Eq. (1.49), we have that ORNI - AEC - OFFICIAL 2ul NS Ng Feu Peep : (1.50) m2c4R2 On expressing No in terms of the density of the sample, we have that that . I 2 at. wt () (1.51) m2c4 RZI mor where N. is Avogadro's number, o the density of the sample and at, wt is www ws A the atomic weight or average atomic weight for a sample containing more m. than one kind of atom. Using Eq. (1.51), we are now able to convert our measured intens ity I into absolute units after determining P, since all other quantities are known or obtainable from the experiment.* 2.1.2 Measurement of the Power in the Incident Beam Using the Warren monochromator with ordinary commercial x-ray diffraction tubes, Po will be of the order of 108 counts per second for a beam divergence of about 2. Hence P. is not measurable directly with *For other geometries see the International Tables for X-ray Crystallography, Vol. II, 291 (1959). INL - AEC - OFFICIAL present-day x-ray counting equipment where dead time corrections become appreciable above 204 counts per second. Even the use of the multiple- foil technique would require an extrapolation over 4 to 5 orders of magnitude. The most widely used technique for measuring P, in diffuse ORXL - AEC - OFFICIAL X-ray scattering experiments is to measure the intensity scattered from an amorphous material substituted for the sample, One advantage of using an amorphous scatterer is that at high diffraction angles the atoms do not diffract cooperatively, so the intensity approaches that for a random assemblage of atoms. Also, it eliminates having to measure the area of the receiving slit if the same slit is used for the sample measurements. Following the same procedure as that used in deriving Eq. (1.51), we may express the intensity in electron units per molecule for our amorphous standard as denoting by prime those values peculiar to the amorphous scatterer. If we knot can then be determined by simply replacing our sample with an amorphous scatter and measuring its scattered intensity at only one position. A motivation for using an amorphous material is the ease of calculating which is expressed as the sum of the coherent plus incoherent scattering, ( ) - se to ne coh 12 + 2 , (1:53) (1.53) B2 after James (30), where f is the coherent atomic scattering factor, z the atomic number, the incoherent scattering function, and B2 the Breit-Dirac reccil correction factor for a detector counting individual vi mwengu wote wanitama quanta. ORNL - AEC - OFFICIAL in handen when comes to the brand ---..-- . . . The accuracy of the theoretical values of the atomic scattering factors . ORNL - AEC - OFFICIAL and especially the incoherent scattering factors is questionable in view of recent experimental results (31, 32, 33). In addition, the ass wmption has been made that in using an amorphous scatter the atoms scatter completely Independently of one another at reasonably high diffraction angles. This assumption is not completely valid but may be somewhat compensated l'or by measuring the scattered intensity out to the limiting sphere of reflection and determining the average value of the intensity by matching the area under the experimental curve with the smoothed curve as shown in Fig. 8. One chooses a high angle of scattering for the amorphous scatterer where the intensity corresponds to the average value. In calculating the value of Ieu per molecule for polystyrene using Cais as its molecule, we choose to make the calculation for sino/n = 0.5 A-1, 100° 20 for Cu Ka radiation where the average curve intersects the measured intensity. Using what are considered the most accurate theoretical values of the atomic scattering factors (34) as given by Freeman (35) for coherent scattering from carbon, by Hanson et al. (36) for hydrogen and incoherent scattering by Keating and Vinegard (37) for carbon and Compton and Allison (38) for hydrogen, we have . م. - :ممممممممممسه للمسخممتدهمه میدونه من سنه حته ينشئولی ن سونجمنه the B2 .82 [ 18 + s + 12 =9)6(2-3)1] (.54) - 8[(1.685)2 + (0.071)2 + 1.65* + 1.037] = 64.7. We have seen that this calculation involves several assumptions. So, for the amorphous scatterer to be a truly absolute standard for RNL - AEC - OFFICIAL 36 measuring Por Cu should be determined experimentally using a direct approach with no assumptions. This of course involves a direct measurement ORNI - AEC - OFFICIAL of Poo Following the suggestion of Batterman, Chipman, and DeMarco (31). a nearly perfect single crystal of silicon was used to scale the power of the direct beam down to within the linear range of the counting equipment. This method is based on the fact that a nearly perfect crystal will diffract at the Bragg angle only those x-rays that are incident within a few seconds of arc of this angle. Hence, if the incident beam has a divergence of about 1° and the width of the Bragg reflection is only a few seconds of arc, there is about a factor of 103 reduction in power. Thus the power of the integrated reflections from the nearly perfect silicon crystal can be measured without exceeding the linear range of our counting system. This counting rate was no more i than 4500 counts per second with a dead time loss less than 1% for our equipment using a scintillation counter. So P. was limited to about 105 counts per second for this experiment. Then the incident beam power is reduced so that it may be counted directly by using a very small hole in a lead sheet. The silicon crystal is again rocked through the reflecting range with the incident beam power reduced. The value of P. then is just the ratio of the integrated intensities times the count rate of the direct beam through the pinhole. This method does not depend on the incident beam having a wiform intensity but does depend on the diffracting efficiency of the silicon crystal being uniform over the area irradiated. This was checked by irradiating a different spot on the crystal for each of four measurements stiveria ORNL - AEC - OFFICIAL 37 and found to vary by less than 11%. The sensitive area of the scintillation counted was determined to be uniform within 11% over an area of 1 cma, larger than necessary to intercept the entire diffracted beam from the crystal. With this technique, it is possible to measure P. to +1% or ORNL - AEC - OFFICIAL better. With the measured Po, the scattered intensity from several samples of commercially available extruded ASTM D-703 type 1 polystyrene rod was determined at 100* 20 using LiF monochromated Cu Ka radiation. All values checked within 41% of the mean value. Several circular hole receiving slits of different area were used. The distance R from the irradiated volume of the sample to the receiving slit was measured directly and by varying the distance from the sample to the receiving slit. These numbers checked to within 11/3%. The only other unknown is the mass absorption coefficient (). Using samples of polystyrene with thicknesses greater than 1/2 in, and known to £0.0002 in., the linear absorption coefficient was determined by using the known P. incident on the sample and measuring the transmitted power. With a scintillation counter and pulse-height discrimination, the purity of the transmitted . . beam was checked and no measurable shift in pulse he::ght occurred because . . W YEAH of preferential absorption of the softer components of the beam. With the measured density of 1.0492 10,0005 g/cms, the mass absorption coefficient of polystyrene was found to be 4.027 cm2/8 for Cu Ka radiation. This value agrees well with the value of 4.005 cm2/8 determined by Batterman (39) using the multiple-absorber technique. These measured values were used in Eq. (1.52) with the result that I, per CHE molecule of polystyrene is 61.1 10.6 at 100° 20 using Cu Ka radiation, RNL - AEC - OFFICIAL This value is 6% smaller than that calculated from theory. The errors ORHLAEC - OF resulting from the use of the wrong values for converting the intensity to absolute units will be discussed in the next section. It is now much easier to measure the power of the incident beams using a calibrated amorphous scatterer as the standard than to repeat the integrated intensity, measurements using a nearly perfect single : crystal. The final equation for converting the measured power in counts per second from our sample to absolute units is obtained on eliminating P. from Eq. (1.51) by dividing by Eq. (1.52) giving I at. wt eu I. I at. wt (De teus 13_(euy. Feu ' 'I'mol. wt ()'p (1.55) 2.2 Corrections for Background Intensities It is important that unwanted contributions to the intensity be reduced as much as possible as the Laue monotonic scattering is weak and larger uncertainties can be introduced. For this reason the use of a monochromator is necessary to eliminate fluorescence of the sample by the short-wavelength spectrum from the x-ray source. In addition, - - - -- we will want to convert the measured intensities to absolute units . . - - . . . . . requiring the removal of the white radiation so that the wavelength spread is small. We will assume that the incident x-ray source is . . . . . . . . . - from a single-crystal monochromator for this discussion. The corrections - i - ' . 1 . i for the unwanted contributions to the measured intensity will be discussed in the order in which we might normally correct for them and not in order of their importance. The total measured intensity is the sum of the following contributions: Iorder, the modulated Laue monotonic intensity we ORNL - AEC - OFFICIAL seek to recover; Iys, the fluorescence from the sample and diffraction by submultiple wavelength components of the beam passed by the monochromator; Ip, scattering of the x-ray beam by objects other than the sample; In, Compton rödified or incoherent scattering from the sample; Inps, temperature-diffuse scattering from the sample: Ifund. sharp fundamental Bragg reflections, and Iy, Huang diffuse intensity. ORNI - ALC - OFFICIAL 2.2.1 Fluorescence and Submultiple Wavelength Diffraction Even with a monochromator, the 12 and lower-order harmonics of the characteristic radiation from the x-ray tube will be diffracted by the higher order reflections at the same 2e setting of the monochromator crystal. Because of photoelectric absorption in the sample, fluorescent radiation will be superimposed on the diffracted beam with a wavelength close enough to the characteristic radiation - that they are counted. This causes a uniform increase in the background intensity. In addition, the 1/2 radiation will be diffracted by the sample at a position corresponding to twice the d spacing for diffraction with the characteristic .. For example, the x-rays of one-ball the wavelength of the characteristic radiation will be diffracted by the 200 planes at the position corresponding to diffraction by the 100 planes of the characteristic wavelength. Thus, unless our detecting system has sufficient resolution to discriminate against 12 radiation, a Bragg peak of 1/2 radiation will occur at the super- structure positions which are areas of interest in order studies. There- - - - fore Na components must be removed from the measured intensity. ORNL - AEC - OFFICIAL . 40 ORNI - AEC - OFFICIAL Operation of the x-ray tube at a voltage lower than the excitation voltage for N2 radiation works well without sacrificing too much intensity for targets of molybdenum and higher in atomic number. However, one must still contend with fluorescence. The use of Ross balanced filters (40) will permit elimination of both the submultiple Wavelengths intensity peaks and fluorescence from the measured Intensity. A filter pair is chosen such that their K absorption edges bracket as close as possible the Ka radiation in use. A calculation is then made to determine the thickness of one of the filters for maximum intensity difference. The other filter thickness is calculated to match this absorption for 1/2. The filters are experimentally checked and, if necessary, corrected for exact balance. This check is easily made using a single-crystal sample oriented for strong diffraction of the characteristic radiation for a first order reflection then balancing the filters until the intensity difference leaves no peak but a smoothly varying intensity through the position corresponding to twice the a spacing. Since we are taking the difference between two measurements of the intensity, the cosmic-ray and instrument contributions to the background.count are also removed at this time. This balanced filter pair must be placed in the incident beam to remove all the fluorescence, Fluorescent radiation of or near the same wavelength as the characteristic would still be passed with the filters in the diffracted beam.. Because the difference between two intensity measurements have to be taken, we more than double the experimental time required to obtain the data. The use of monochromator crystals with no second-order reflection might prove sufficient in eliminating the need for Ross balanced flīters if the intensity sacrifice is not too great. ORNL - AEC - OFFICIAL ishirini narednom . Parasitic Scattering ORNL-AIC-OFFICIAL Any objects through which the x-ray beam passes on traveling from the monochromator crystal to the receiving slit are sources of scattered x rays which contribute to unwanted background intensities. - - from these sources is a scatter slit in front of the receiving silit. : If the entire receiving slit "sees" only the volume of the sample irradiated by the beam, we have eliminated all but air scatter as a possible parasitic source of background radiation. Air scatter is eliminated either by evacuating the volume near the sample or by using a He or Hą atmosphere. The air scatter from the diffracted bean is usually of no consequence. The scatter slit in front of the receiving slit is best located * II T. , tan as near the sample as possible. Since the beam spread on the semiple is largest at low 20, the area of the scatter slit is adjusted at the wiwi -.-.-- lowest 20 of interest, to allow the receiving slit to intercept x rays coming from any part of the irradiated volume of the sample. It' the sample is surrounded by a vacuum shield or any other shield, it is very important that the receiving slit not "see" the area where the incident beam traverses this shield. - - - - - - Wi- experimentally using a beam trap in place of the sample and applying & geometrical factor dependent on 20. However, et 20 below about 15• this correction is not reliable and could cause serious error. 2.2.3 Compton Modified Scattering The intensity from Compton scattering is considered to be without structure and increases smoothly with increasing scattering ; ORNL - AEC - OFFICIAL angle. There is usually no large uncertainties introduced in correcting ORNI - AIC-OISICIAL for this contribution to the measured intensity. After having taken the balanced filter difference as the measured intensity, we would now convert these measurements to intensities in absolute units as described previously. The Compton intensity would then be computed from the theoretically known incoherent scattering factors and subtracted. The average intensity in electron units per atom for the Lave TO monotonic diffuse scattering is X,X (f- fp), while for Compton scattering it is (XA(Z - 3) A + Xng (Z. -3). A comparison of these two intensities for an alloy of Cu + 16 at. % Al at sin 0/2 = 0.3 gives 19.3 electron wits per aton for the Laue monotonic intensity and about 7.6 for the Compton intensity. Thus if the error in the atomic incoherent scattering factors is about 6% as might be presumed to be the case from the previous measurements for polystyrene, the remaining error in the Laue monotonic is only about 2%. In more dilute alloys or where the difference in atomic scattering factors are less favorable, the Compton intensity can be as large as the Laue monotonic. In this case any multiplication error in converting the measured intensity into absolute units will be doubled since we are subtracting either too much or too little Compton intensity by just the multiplication error. Thus, with strict attention given to measuring it correc:ly the power of the incident beam, to eliminate a multiplication error, correcting for the Compton intensity should not be a source of appreciable error. ORN1-AEC - OSSICIAL ORNI - AEC - OFFICIAL 32.7.4. Temperature-Diffuse Scattering The theoretical treatment of temperature-diffuse scattering has been confined mostly to either monatomic or ordered lattices. Experi- mental measurements of x-ray diffuse scattering associated with the thermal vibration of the atoms generally confirms the validity of this treatment. Janos (431) has given a rather thorough discussion of temperature-diffuse scattering and we intend only to point out means of correcting for its contribution to the unwanted background scattering. Even if an adequate theory exists to calculate the distribution of intensity associated with the thermal motion of the atoms in un alloy, we would need to know the vibrational frequency spectrum of the crystal. This frequency spectrum is not known theoretically nor are the atomic force constants from which it can te calculated. We are left then with having to experimentally determine the temperature-diffuse intensity so that it can be eliminated. Without a specific theory for solid solutions, it will be assumed that to a first approximation the temperature dependence of the temperature-diffuse scattering from an alloy with order behaves in much the same way as that from monatomic cubic solids. It is the temperature dependence of the thermal-diffuse scattering which provides & method for its removal from the temperature independent local-order scattering. The local-order scattering is considered temperature- independent at temperatures below that at which no appreciable change in local order occurs. The intensity of thermal-diffuse scattering is proportional to the energy of the elastic waves which describes the atomic motion. With the measurements of intensity being made at temperatures T high compared to the Debye temperature, this enerw is ORNL - AEC - OFFICIAL very nearly kr for each wave where k is the Boltzmann constant. Tous ORNI - AEC - OFFICIAL the temperature-diffuse intensity is assumed to vary linearly with temperature. By, making measurements at two temperatures, the temperature dependent part of the intensity can be determined. With reference to Fig. 9, the temperature independent diffuse intensity In is given with the temperature in degrees Kelvin. It is probably better to make the measurements at room temperature and below even though the - . - - approximation that E = kr is less valid for TN . [Hq 0 H3]> . Fig.to ORHL-LEC - OFFICIAL 0311* 1.CC - OFFICIAL ORNL-DWG 65-771 t ..- ... L ; - - - Fig.1/ OG?![25C - OFFICIAL ORIFL-ALC - OFFICH voisiv-98-1:20 Donwide.. - ir u. . .. 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