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X-ray diffraction of modulated structures
X-RAY TOKt’ZOU
DIFFRACTION OF MODULATED STRUCTURES
TSUJIMOTO, KEhXI HASHIXIOTO and KAZUO SAITO
National Research Institute for Metals, Meguro-ku, Tokyo tj?, Japan (Rewired
15 November 1974: in rerised &mn 31 June 1976)
Abstract-Modulated structures are regarded as a macrolattice. in which each lattice point is a ‘unit region of concentration variation’ (abbreviated to ‘unit region’) and an average lattice parameter is Qa,,. The X-ray intensity diffracted from modulated structures. I,(s), is given by J,(s) = I.(s)‘f&i. I,(s) is the scattering intensity from the ‘unit region’ and I,,(s) is that from the imaginary macrolattice in vvhich each lattice point is the unit scatterin g factor. A region containing one zone-complex has been chosen as the ‘unit region’. and the effects of the lattice spacing of the macrolattice and its disturbance on diffraction patterns have been discussed. When zone-complexes are small. Ids-) is dominated by J,(s) and peaks of I,(s) are observed as side-bands or additional diffraction lines. The number of additional diffraction lines increases with an increase in Qae and typical side-bands appear at a stage of this process. When the periodicity of the macroiattice is disturbed deeply or Qa,, is large, fU(s) is observed directly as diffraction effects. The present theory explains reasonably the asymmetry of side-bands in position, the movement of main diffraction line with aging and the change of side-bands into diRraction lines of metastable phases during aging. R&urn&On co&dire les structures modulees comme un macroriseau dont chaque noeud est une “maille de variation de concentration” (abregee en “maille” dans la suite du texte). et ou Qa, est un paramttre reticulaire moyen. L’intensite de rayons X I,(s) diffractee par une structure modul5e est donnee par la relation i,(s) = f.(s)*l~s). I,(s) est I’intensite diffractCe par une “maille” et IS(S) est I’intensitt diffractee par Ie macrorbeau imaginaire dont chaque noeud est le facteur de diffusion unite. On a choisi une region contenant une zone complexe comme “maille” et on a Ctudie les effers du parametre du macroreseau et de les perturbations sur les diagrammes de diffraction. Quand les zones sont petites, I,(s) est dominee par I&) et on observe les pits de I,(s) sous forme de bandes laterales ou de raies de diffraction supplimentaires. Le nombre de raies de diffraction supplementaires augment lorsqu’augmente Qao et les bandes latirales ne sont qu’une &ape de ce processus. Quand la periodiciti du macroreseau est ptofondement perturbee ou lorsque Quo est grand, on observe direcrement I&) sous forme d’effets de diffraction. Cette thiorie explique raisonnab~ement la position dissyrn& trique des bandes laterales, fe dtplacement des raies de diffraction principales au tours du vi~iliissement et la transforma~on des bandes laterales en raies de diffraction de phases metastables au tours du vieillissement. Zusammenfassung-Modulierte Strukturen werden als ein Makrogitter betrachtet, in dem jeder Gitterpunkt ein “Einheitsgebiet einer Konzentrationsvariation” (abgekiirzt “Einheitsgebiet”) ist; der mittIerc Gitterparameter ist Qao Die von modulierten Strukturen reflektierte Riintgenstrahlintensitlt 1,(s) ist gegeben durch I,(s) = I&)*1,&). I,(s) ist die Streuintensitlt des “Einheitsgebietes”, I,,(s) ist diejenige des imaginlren Makrogitters, in welchem jeder Gitterpunkt der Einheitsstreufaktor ist. Als “Einheitspebiet” wird ein Gebiet mit einem Zonen-Komplex gewiihlt. Es wurden die Einfliisse des Gitterabstandes im Uakrogitter und dessen Stiirung auf die Beugungsdiagramme diskutiert. Sind die Zonen-Komplexe klein. wird i,(s) von I,(s) bestimmt; es werden Maxima von I,(s) als Seitenbgnder oder zusstziiche Beugungslinien beobachtet. Die Zahi zudtzlicher Beugungstinien nimmt mit zunehmendem Gitterparameter Oa, zu; typische Seitenblnder wird auf ein Zwischenstadium dieses Prozesses beobachtet. 1st die Per&dititat des Makrogitters stark gestort, oder ist Qao grol3. dann wird I,(s) direkt in der Beugung beobachtet. Die vorgelegte Theorie erkiPrt zuf~edenstellend die Asymmetrie der Seitenblnder in ihrer Position, die Bewegung der Hauptbeu~ungslinie mit der Alterung und den Wechsel der Seitenb%tder in Beugungslinien metastabiler Phasen wiihrend des Alterns.
I. INTRODUCTION
of mirror reflection across the main diffraction line to the low-angle side-band. The side-bands change The formation of modulated structures in aged into fixed diffraction lines during aging. The main difCu+!/,Ti and Ni-lO”/;;Ti alloys has been verified by fraction line moves to a high angle side with aging means of X-ray diffraction and electron microstime, after a certain stage was reached which is detercopy [l-15]. On the diffraction patterns of these mined by aging temperatures. The X-ray diffraction alloys the following are noted. The side-bands are theory of modulated structures must be able to broad and asymmetric in position, intensity and half- explain these phenomena on appropriate solute conwidth: the high-angle sideband is not in the relation centration variation models. 295 A..\,. 253-4
‘96
TSGJIMOTO,
HASHIMOTO
AND
In modulated structures a local concentration variation or a ‘unit region of concentration variation’ (abbreviated to a ‘unit region’) is repeated at semiregular intervals. The network of ‘unit regions’ can be regarded as a macrolattice. in which each lattice point is the ‘unit region’. In the present paper, the effects of the lattice spacing of the macrolattice and its disturbance on diffraction patterns have been discussed. A small region of solid solution containing one zone-complex has been chosen as the ‘unit region’ to explain the concept of the present theory and to depict the essential features of the diffraction patterns from modulated structures. Although the treatment is simple, this theory gives a general view of various aspects of diffraction effects by modulated structures and makes it possible to explain varieties of sidebands. Finally, diffraction patterns of aged Cu-4”,;Ti and Ni-lO%Ti alloys have been discussed. Up to this time various solid solution models with solute concentration variation have been proposed [16-281. However, most of them have not given any explanation for the asymmetry in position of sidebands. Some researchers explained this asymmetry by assuming a random distribution of zone-complexes E23.241. However, this concept does not agree with the observation of periodic microstructures in sidebands alloys. Also, behavior of side-bands at the later stage of aging have not been discussed in all previous papers. De Fontaine simulated diffraction patterns from modulated structures by using an electronic device [25]. The present paper gives a clear explanation based on an acceptable physical mode1 for some of his results. The germ for the concept of macrolattice can be seen in some papers [Z, 21,26,28-j. In this paper this conception is identified clearly and is used completely. Although the present theory is described, for simplicity, for one dimensional model except section 11.2.4, it can be easily extended to three-dimensional model. We have only to calculate in the same way the scattering intensity from a three-dimensional ‘unit region’ and a three-dimensional macrolattice, instead of one-dimensional treatment. II. THEORY 2.1 Fandamentals~or
di~~uctjon by adulated
structure
The amplitude of X-ray diffracted from a crystal lattice, A(s), is given by no-1 A(s) = C fn(s)-exp(-2xisr,), (1) n=O in which s = 2.sin B/E.,@is a diffraction angle, and j. is a characteristic wavelength of radiation. f”(s) is a scattering factor of an atom at the nth lattice point whose coordinate is r,. If the effect of the variation of scattering factors is markedly smaller than that of atomic spacings, the equation (1) may be replaced by A(s) = fe (s) f 2 exp( - 2nisrJ, n=O
(2)
SAITO:
X-RAY DIFFRACTION
fo(s) is an average scattering factor of atoms in an alloy. The present treatment for diffraction of the modulated structure is based on the conception of the macrolattice. The scattering amplitude from the ‘unit region’, A,(s). is calculated using the equation (1) or (2). Then the total amplitude of X-ray diffracted from the macrolattice .4,(s), may be written as x0-
A,(s) = A,(s)* y
1
expj-%isR,),
(3)
,G‘O
in which R,v is the coordinate of the IVth lattice point of the macrolattice. When the lattice parameter of the macrolattice is Quo, .s,,A,(s)
=
=
x
1
v =0
exp( - h’sR,)
sin asiV, Qae sin nsQn e
+exp {nisQa,(.Y,
- 1)).
Here, a0 is an average atomic spacing of the alloy. The totaf intensity of X-ray diffracted from the macrolattice, I{(s), is given by I,(s) = f”(S) x I,(s).
(4)
I,(s) = ‘4,(s)*Ads)*
(3
Here,
and Ids) = AAs). A&)* = (sin nsNoQaO,‘sin ;isQae)‘. (6) f&s) may be regarded as the intensity of X-ray diffracted from the imaginary macrolattice, in which each lattice point is not the ‘unit region’ but the unit scattering factor. I,(s) is a periodic function and gives sharp peaks at positions of s = p/Quo. p is an arbitrary integer. In the modulated structure, however. *unit regions’ are not always distributed with perfect or sharp periodicity as atoms in a usual crystal lattice. Or rather, they are distributed in an intermediate state between perfect order and perfect random. Here. we define I&) as intensity of X-ray diffracted from the imaginary macrolattice with diffuse periodicity. Although the exact function form of ids) is unknown. I&) is considered to give diffuse peaks at the same positions as f,(s). We suppose that peaks of I&) become broader with increasing disturbance of periodicity (see Appendix). By replacing I,(s) by I,(s). the expression for the X-ray intensity diffracted from the modulated structure, I,(s), is obtained : (7) I,(s) = I,(s) x rdH. Two features of the diffraction theory for the modulated structure are noted. One is diffuse periodicity in the arrangement of lattice points of the macrolattice. This feature brings about diffuse diffraction intensity. The other is that the scattering factor of lattice points of the macrolattice is strongly dependent
TSUJI?vfOTO. HASHMOTO
A4‘D
S‘AIT0:
X-RAY DIFFRACTION
797
upon scattering angles in a complicated manner. This is caused by the fact that ,4,(s) itself comes from diffraction of a large group of atoms. Z.1 Scarcering tion pcirrern
intensity from ‘unit region’ and difrac-
Examples of solute concentration variation models are shown in Figs. 1 and 2. Here, solute concentration variations of Fig. 1 are perfectly periodic and those of Fig. 1 are semi-periodic with an average spacing QL%. If a wavelength of sinusoidal concentration variation. Quo. is taken as the ‘unit region’ as in Fig. I(a). I,(s) and I,(s) will have peaks at the positions of 11~1~I: p, Qno. Thus. I,,,(S)and side-bands are symmetrical in positions about the main diffraction line. In the cases of Figs. 1 (b) and (c). any region containing a wayslength of concentration variation, Quo, may be chosen as the ‘unit region’. In such cases peaks of I,(s) are at the positions of h/a,, 2 piQao, but those of I,(s) are generally not. However, as peaks of IAs) are extremely sharp in comp~ison with those of I,(S), we get pcdks of I,(S) at positions of I,(s) after all. It must be noted that side-bands arising from any perfectly periodic concentration variations are symmetrical in positions about the main diffraction line and that they are as sharp as normal diffraction lines. Local concentration variations in Fig. 2 are related to the configuration of G.P. zone. (a) corresponds to a 3-phase model and (b) to a 2-phase model. When atomic spacings are a, in the range from &I; to n&b + 11~~7~ as in Fig. 2(b) and the average atomic scattering factor is fl. the scattering amplitude from this part. il$), is given by A2(s)
=
f? . s’n aStlJnl sin xsffI x exp(-ZKisnbnb).exp:nis~~(n~
- 1);.
If any parts of the ‘unit region’ do not have a close value of atomic spacings, the scattering wave from the part of atomic spacings LZ?will not interact with those from other parts. Then, IL(s) = A3(s)*AI(s)* = ,‘i *[sin nsrlzn,/sin ASPS)‘.Hence. I,(s) for Figs. Z(a)
-X
Fig. 1. Periodic solute concentration variation models.
Fig. 2. Local solute distribution and ‘unit region’ in semiperiodic concentration variation models.
and (b) may be summarized as follows: Each part gives the intensity f’ -(sin nsna;‘sin nsn)’ in which n and f are the number of atoms of the part having an atomic spacing a and the average atomic scattering factor of that part, and I,(s) is the sum total of the intensity diffracted from each part. f’ (sin ~snccjsin RSFZ& gives peaks of maximum intensity 6j2 and halfwidth I;nu at positions of s = q!u. 4 is an arbitrary integer. The peaks become stronger and sharper with an increase in n. ~i~raction patterns arising from the modulated structures having the -unit regions’ shown in Fig. 2(a) and (b) are shown in Figs. 3 and 1. respectively. For the calculation of I&s) equation (2) was used and Id(s) were considered to be those as indicated by chained lines. I,(s) are presented by broad lines or by hatched areas. It is noted that the main diffraction line or the high-angle side-band is weak or not observable for the concentration variation of Fig. 2(b). In the present theory ail kinds of local concentration variations may be adopted as a ‘unit region’. In the following explanation, however, the theory will be described for convenience on the basis of the 3-phase model of G.P.-zone in a semi-periodic array. This model can explain most reasonably the broadness and the asymmetry in position of side-bands, the existence and the movement of a main diffraction line with aging, and the change of side-bands into fixed diffraction lines. 2.3 Some specific aspects motiulnred structures
of diffraction
eficrs
from
Modutated structures are formed during aging aft.er quenching. We choose a region containing one zonrcomplex as the ‘unit region’. The size of the zonecomplexes and the distance between them increase with the progress of aging. We do not obtain enough information concerning the relationship between aging time and degree of periodicity in the distribution of zone-complexes. Ardell and Nicholson described that the array of ‘unit regions’ in a Ni-Al alloy is at random at the early stage of aging and becomes regular with aging time [303. On the other hand, the array will be relatively regular from the early stage of aging if an alloy decomposes spinodaily. This problem is related to the origin of alignment of decomposition products and decomposition mechanism.
-----+
Intensity
,pp
v, .;-+----.-_.>8p - ~-~_~+~_._.A--’ i !
\ \\ :I
-3.8 ‘1 +._.-\
:\ \ ---.-__
a-
-
943 s
I-
\
&I- I
+_
/
:i
I
-
rd‘
g
8
f&j g
F
a
: -*Intensity =i ___._.._..__________________ 0
St-
3
3-
1 -
‘\ ;
2 tl
‘4;_~___--_____-__‘__._ x __._ ___ __\_
:“-. . .._
I f_____ u
i
:\ : :, ! ~~,+-.-._._._._._q-
cn,
TSt’JIMOfO.
HASHI~OT~
.VGDSAITO:
X-RAY DIFFRACTI04‘
299
Fig. 4. X-ray intensity diffracted from clusters as in Fig. I(b) in a periodic array. a2 = 3.675 A, a; = 3.615 A and
Q = 138 are assumed. The effect of the degree of periodicity in the distribution of zone-complexes on diffraction patterns is shown in Fig. 3, in which (a) represents the diffraction pattern from zone-complexes in sharply periodic array and {b) and (c) represent those from zone-complexes in semi-periodic array. With increasing diffuseness of periodicity. each peak in 1&s)becomes broader and, as a consequence, the side-bands become broader and more asymmetric. In the case of a nearly random distribution, Cd),the foot of each peak of Id(s) falls on the foot of the nearest peak so that Id(s) becomes nearly constant. In this case, I,(s) can be determined directly from the diffraction pattern. The change in symmetry of side-bands corresponds to the change of the dominating factor on diffraction from f,(s) to Us). Figure 5 shows the effect of the distance Qao between the nearest zone-complexes on diffraction patterns. Here, 1&s) is assumed, for convenience, to have sharp peaks. Except the main diffraction line, one additional diffraction line is observed in (a), three lines in (b), and many lines in (c). It is clear that jj the number of additional diffraction lines increases 11 as Q increases and that typical side-bands, in which one additional diffraction line is observed on each iI j ! side of the main diffraction tine, are only a stage in i! the increasing process of the number of additiona 1 ! diffraction lines. In the case of a large Q value, (d), i! additional diffraction lines are so many and close that i1 they do not give virtually any effects on the diffraction i! pattern and the same diffraction effect as I,(s) is j! observed. i ! increase or decrease in solute concentration of each d zone of the zone-complex is reflected in the position of ‘peaks of I,(s). Increase in the size of the zonecomplex is reflected in the sharpness of peaks of I,(s). For small zone-complexes additional diffraction -s lines are observed in a wide range of Q values, i.e. Fig. 5. X-ray intensity diffracted from zone-complexes in from the early stage of aging to the later stage. For a periodic_array. The-same atomic spacings as in Fig. 3. large zone-compieaes additional diffraction lines are Q are 69 in (a). IiS in (b), 212 in fc) and 917 in (d).
300
TSUJIMOTO,
H~SHI~lOTO
AND
observed in a narrow range of Q values, only in the relatively larer stage of aging and they become strong rapidly during short time of aging. There will be various combinations of the size of zone-complexes. the distance between them and the degree of periodicity. If the periodicity in Fig. S(c) were diffuse, the feet of the peaks of I,(s) would overlap each other and I,(s) would become nearly constant. In such case I,(s) would be observed as the diffraction pattern. If the periodicity in Fig. j(b) were diffuse, the two peaks on the low-angle side would join together and a broad diffraction band would be observed. Although the asymmetry of the side-bands is only in intensity in Fig. 3(a), it is in position and in intensity in Figs. 3(b) and (c). This means that, for the side-bands showing the asymmetry in position, it is not appropriate to assume perfectly periodic array of zone-complexes. Figures 3 and 5 are drawn in the case of c1 - ca = 2 (q, - ct). In those circumstances the asymmetry of the side-bands increases with the increase in Q. This agrees with the experimental results described Iater. 2.4 ~l~re~t~ur~o~ dijj%action lines
between
side-bands
and
~~r~ul
The total intensity diffracted from the modulated structure is given by the equation (7). In extreme cases, the positions of diffraction lines are determined by the sharper peak of the two functions, Id(s) and I,(s). Usually, the dominating function changes gradually and continuously from I,(s) to I&) or vice versa. Thus. it is necessary to determine which function is dominant on a diffraction pattern. We define the patterns dominated by r&s) as side-bands and those dominated by i&) as normal diffraction lines. The reciprocal lattice points (abbreviated to relpoints) for the solid solution of the average atomic spacing a0 are indicated by the indices hkl. The reciprocal lattice of the macrolattice (abb. to sub-relattice) has the lattice spacing l/Quo and some of the subtelattice points (abb. to sub-relpoints) coincide with the hkl relpoints. Sub-relpoints around a relpoint are indexed as h’k’l’ by considering the hkl relpoint to be the origin of the sub-relattice. An hkl relpoint is observed as the main diffraction line at the scattering angle 8 on a powder X-ray diffraction pattern and an h’k’l’ sub-reIpo~t belonging to the hkl relpoint is observed as the diffraction intensity at the scattering angle 0 + AB1. Then, Q and AOl is related by
SAITO:
X-RAY DIFFRrtCTfON
a zone is a + Aa and that of a matrix is a, the diffraction line of the zone at the scattering angle 8 + A-\8a and that of the matrix at the angle 6 is related by AtJJtan 0 = Ann = constant.
When a region containing one zone-complex and matrix is chosen as the ‘unit region’ and the atomic spacings of inner- and outer-zone are n, and a,, the equation (9) holds for each zone of zone-complex. In the present model the reciprocal lattice of the ‘unit region’ consists of the relpoints for each zone of zone-complex and the matrix. Diffraction effects are observable only when a sub-relpoint coincides with the reciprocal lattice points of the *unit region’. All the additional diffraction lines except the main diffraction line are related to the relpoints of each zone of zone-complex. We can expect at the early stage of aging that the relpoints for the zone-complex are diffuse and cover some sub-relpoints and, as a consequence, the side-bands are frequently observed. The relpoints for the zone-complex become sharp at the later stage of aging. In this case we can observe the additional lines, only when the sub-relpoints are diffuse or dense. Some of the diffraction patterns have an intermediate character between side-bands and the diffraction lines of zone-complex. That is, the intensity peaks of I,(s) are between the peaks of I,(s) and those of f&s). The diffraction patterns according to the equations (8) and (9) are compared in Fig. 6. In the case of ci - co > co - ca, the peaks of I,(s) on the highangle side are closer to the main diffraction lines than those on the low-angle side. This brings about the asymmetry in the positions of the peaks of I,(s). A& becomes larger with the increase in Miller indices and this brings about the dependence of Q on Miller indices. Thus the values of Q estiniated from the sidebands of high indices have a tendency to be small. (200) side-bands are in a special situation. A62 of them, especially one on the high-angle side, are markedly smaller than ABi, and hence the estimated value for Q tends to be large. (113) side-bands observed usually correspond to h = 3, hence the estimated values of Q is not so small as anticipated in the case ofh=l.
(2201 (311)
which is reduced to the formula well known for sidebands
(222)
when h’ = 1 and k’, 1’ = 0. On the other hand, I,(s) is the function of the same type with the diffraction effect from ‘unit regions’ at a random distribution. When the atomic spacing of
I,1 II/ I l&l
,ll/!I
(111) 000~
Q = (hh’ + kk’ + ll’)*tan6/(na + kZ + 1’).A6, (8)
Q = h*tan 0/(h’ + k? -t 12)*A0,
(9)
1
9,
I
1
I ti
.-
-S Fig. 6. Comparison between side-bands or IAs) (solid Lines) and diffraction lines of zone-complexes or I,(s) (broken lines). The same atomic spacings as in Fig. 3 and Q = 40 are assumed. Id(s) is given for h’ = 1 and k’. 1’ = 0 except (113), for which h’ = 1 and k’, 1’ = 0 (far from M) and I’ = 1 and h’, k’ = 0 are given.
TSUJIMOTO,
HXSHIMOTO
III. EXPERIMESTAL RESULTS DISCUSSIOS
Ah-D
AXD
Diffraction patterns of the Cu-P~Ti alloy and the Xi-10aOTi alloy have been examined using a Cukr, radiation and a Guinier type camera. The side-bands of the Cu39;Ti alloy are shown in Fig. 7. The side-bands are weak. diffuse, relatively symmetric and in a large separation from the main diffraction line at the early stage of aging. They approach the main diffraction line with the progress of aging and become intense. sharp and asymmetric. The high-angle side-band is always intenser, sharper and nearer to the main diffraction line than the low-
SAITO:
X-RAY DIFFRACTIOI\;
301
angle side-band. At a certain stage the low-angle sideband broadens suddenly and then the side-bands are replaced by fixed diffraction lines. The main diffraction line moves to a high angle side from the beginning stage of a_tig at temperatures over 43C. but it begins to move at the later stage of aging at temperatures under 43’C. The main diffraction line keeps moving until it reaches the fixed diffraction line on the high angle side. Intensity of the main diffraction line decreases during aging and this phenomenon is remarkable in the process of the movement of the main diffraction line upon the low temperature aging. We consider that the zone-complexes aggregate during aging and that c2 - c,, > c,, - c2. The broadness of the side-bands represents that the zone-com-
Fig. 7. Side-bands of a Cu--l”/;Ti alloy
302
TSUJIMOTO. HASHIMOTO A>B SAITO:
plexes are in a semi-periodic distribution. The iixreasing asymmetry of the side-bands with aging is attributed to the effect of I,(s), whose peaks become sharper with the increase in the size of the zone-complexes. With the progress of aging, a situation similar to Fig. S(b) is realized. As the periodicity is not so sharp in the alloy, two diffraction lines on the low angle side in Fig. 5(b) are observed as a broad band With further aggregation the situation of Figs. 3(d) or S(d) is realized and I,(s) is directly observed. Thus, the fixed diffraction lines are those arising from the inner- and the outer-zone of the zone-complex and their positions correspond to the atomic spacings of each zone. As the inner- and the outer-zone are in a metastable equilibrium, they are named xl-phase and x,-phase, respectively. The movement of the main diffraction line to the high angle side represents the decrease of solute concentration of matrix. The coincidence of the main diffraction line with the a,-phase diffraction line means the achievement of a 2-phase structure.
X-RAY DIFFR,KITON
The sharpness of the side-bands and the weak asymmetry in their positions may indicate that the distribution of the zone-complexes in the Ni-loll/,-Ii alloy is more periodic than that in the Cu+Ti alloy. The fact that only the moving main diffraction line and the zz-phase line are observable in the course of aging may mean that the transfer from the 3-phase structure to the 2-phase structure takes place at the relatively early stage of aging in this alloy. Thus the patterns in such stage are considered to correspond to Fig. 4. IV. SU~IMARY
The side-bands observed for the Ni-lOF
The network of ‘unit regions’ is regarded as the macrolattice. in which each lattice point is the ‘unit region’ and the average lattice parameter is Que. The X-ray intensity diffracted from modulated structures, I,(s), is given by us) = f&)-l,(s). r=(s) is the scattering intensity from the ‘unit region’ and IAs) is that from the imaginary macrolattice, in which each lattice point is the unit scattering factor. I,(s) gives diffraction peaks at positions of p/a. where a is atomic spacing for each part of the ‘unit region’. I,,(s) gives diffraction peaks at q/Quo. p and q are arbitrary integers. The peaks of I,(s) are diffuse when the size of the part having the same atomic spacing is small. The peaks of Z,,(s)are diffuse when the periodicity of the macrolattice is disturbed. In the present paper a small region containing one zone-complex has been chosen as the ‘unit region+. Effects of the lattice parameter of the macrolattice, its disturbance and the size of the zone-complexes on diffraction patterns have been described. When the peaks of r,(s) are diffuse and those of I&) are sharp, the diffracted intensity I,&) is mainly dominated by I,(s). In this case, the peaks of I,(s) are observed as side-bands or additional diffraction lines. The number of additional diffraction lines increases with an increase in Qao and side-bands are observed at a stage of this process. When the periodicity of the macrolattice is perfect and the zonecomplexes are small, the side-bands have good symmetry in position about the main diffraction line. When the peaks of I&) are diffuse or Quo is large, I,(s) becomes constant and I,(s) is observed directly as the diffraction effects. The effect of the sharpness of the peaks of I,(s) or I,(s) and the effect of the Miller indices of the main diffraction line on diffraction angle of I,(s) have been discussed. The present theory can explain reasonably the following phenomena in the Cu-4% Ti and the Ni-loo/ Ti alloy: the asymmetry of side-bands in position. intensity and halfwidth; the movement of the main diffraction line to the high angle side with aging; the change of sidebands into diffraction lines of the metastable phases during aging.
Fig. 8. Diffraction pattern in the nei~hbourhood of the (200) main diffraction line of a Ni-LO?;Ti alloy plate aged at 600’C for 200 hr.
1. K. Saito, K. Iida and R. Watanabe, J. Japan Inst. Merds 31, 631 (1967) (in Japanese); Eans. Nat. Res. Inst. lwrtais 9, 267 (1%7).
3.2 D~~action
pntterns
of
Ni-lOT
REFERENCES
TSUJIMOTO.
HASHIMOTO
AND SAITO:
2. T. Tsujimoto and K. Saito, Trans. Japan Inst. Metals 17. 5l-t: 537 (1976). 3. K. Saito and R. Watanabe, Japan J. appl. Phys. 8. I-t I 1969). 4. K. Hashimoto and T. Tsujimoto, J. Japan Inst. Metals 39. 436 (1975) (in Japanese); Trans. Nat. Rex Insr. Metals 18. 121 (1976). 5. T. Tsujimoto. J. Japun Inst. Metals 39, 285 (1975) (in Japanese). 6. C. Buckle and J. Xenenc. ,Clem. Sci. Reti. hfet. 57. 435 11960). 7. J. ;\. Comic. A. Datta and W. A. Soffa, &let. Trans. 1. 727 (1973). S. D. E. Lauahlin and J. W. Cahn. Acra .L[et. 23. 329 (19’1;).
-
9. T. Miyazaki, E. Yajima and H. Suga. Trans. Japan Inst. Metah 12. 119 (1971). 10. S. Nishikawa and K. Nagata Trans. Japan Inst. Mera/s 17, 277 (1976). 11. S. Ikeno, S. Saji and S. Hori, J. Japan Inst. Metals 38, 446 (1974) (in Japanese). 12. E. G. Nesterenko and K. V. Chuistov, Fiz. Merei. ,Vferuliooed. 9, 140 (1960). 13. H. Suga T. Miyazaki and E. Yajima, J. Japan Inst. .Clrtals 34, 1190 (1970) (in Japanese). I?. C. Buckle. B. Gentry and J. Manenc, Rec. Met. jl. 147 (1959). 15. D. H. Ben Israel and M. E. Fine, Acta ,Lfet. 11. 1051 (1963). 16. V. Daniel and H. Lipson. Proc. R. Sot. (A) 181. 367 (1943). 17. M. E. Hargreaves. .4cm Cryst. 301, 4 (1951). 18. D. Balli and M. Zakharova. Dokl. Akad. Nauk. USSR 96 453, 737 (195-t). 19. T. J. Tiedema. J. Bouman and W. G. Burgers, Acta Jlrt. 5. 310 (1957). 20. E. Biedermann, tlcta Cryst. 13. 650 (1960). 21. Y. Fukano, J. phys. Sot. Japan 16. 1195 (1961). __. ‘7 M Hillert, M. Cohen and B. L. Averbach, Acta Met. 9. 536 (1961).
X-R.AY DIFFRACTION
303
23. A. Guinier. ;Lcrcr.ifrr. 3. 510 (1955). 2-t. J. Xlancnc. C. r. 4cud. Sci. 248. 1Sll 11959). 25. D. de Fontaine. Locczl .4romic .4rrumqemenrs Sidied b) X-Ruy Dltfktion (edited by J. B. Cohen and J. E. Hilliard). pp. 51-9-t. Gordon & Breach. Sew York (1966). 26. Yu. D. Tiapkin and M. V. Jibuti. .4cra .\frr. 19, 365 (1971). 27. P. M. de Wolff. .4ctn Cr)sr. (A) 30. 777 (1974). 28. H. Bohm, rlcra CrFst. (A) 31, 621 (1975). 29. A. J. Ardell and R. 8. Nicholson, .-lcta ,Lfer. 11, 1295 11966). APPENDIX In order to exolain exoerimental facts. l.,(s) must be a periodic function’which is repeated by a small change of s and has diffuse peaks. From this point of view. we introduced the concept of the macrolattice and the diffuse periodicity as a convenient means. We may suppose the following model as an intermediate state between the perfect periodicity and the perfect randomness. That is. the actual center of the :vth ‘unit region’ is displaced by J. distance Ax,” from the Xth macrolattice point, the probability for large /A.u,~~is lower than that for small I Axu’. and 4”- L 7 A.u, = 0. VT” The displacement ratio to the lattice parametsr. l.x,v, @I,!. can be much larger than that in high temperature crystal. The broadness of the peaks from the macrolattice comes from the fact that in
the maximum value of each term except Ar, = 0 is not concentrated at positions s = p,Qa,. We assume Id(s) as chained lines in Fig. 3.
DIFFRACTION OF MODULATED STRUCTURES
TSUJIMOTO, KEhXI HASHIXIOTO and KAZUO SAITO
National Research Institute for Metals, Meguro-ku, Tokyo tj?, Japan (Rewired
15 November 1974: in rerised &mn 31 June 1976)
Abstract-Modulated structures are regarded as a macrolattice. in which each lattice point is a ‘unit region of concentration variation’ (abbreviated to ‘unit region’) and an average lattice parameter is Qa,,. The X-ray intensity diffracted from modulated structures. I,(s), is given by J,(s) = I.(s)‘f&i. I,(s) is the scattering intensity from the ‘unit region’ and I,,(s) is that from the imaginary macrolattice in vvhich each lattice point is the unit scatterin g factor. A region containing one zone-complex has been chosen as the ‘unit region’. and the effects of the lattice spacing of the macrolattice and its disturbance on diffraction patterns have been discussed. When zone-complexes are small. Ids-) is dominated by J,(s) and peaks of I,(s) are observed as side-bands or additional diffraction lines. The number of additional diffraction lines increases with an increase in Qae and typical side-bands appear at a stage of this process. When the periodicity of the macroiattice is disturbed deeply or Qa,, is large, fU(s) is observed directly as diffraction effects. The present theory explains reasonably the asymmetry of side-bands in position, the movement of main diffraction line with aging and the change of side-bands into diRraction lines of metastable phases during aging. R&urn&On co&dire les structures modulees comme un macroriseau dont chaque noeud est une “maille de variation de concentration” (abregee en “maille” dans la suite du texte). et ou Qa, est un paramttre reticulaire moyen. L’intensite de rayons X I,(s) diffractee par une structure modul5e est donnee par la relation i,(s) = f.(s)*l~s). I,(s) est I’intensite diffractCe par une “maille” et IS(S) est I’intensitt diffractee par Ie macrorbeau imaginaire dont chaque noeud est le facteur de diffusion unite. On a choisi une region contenant une zone complexe comme “maille” et on a Ctudie les effers du parametre du macroreseau et de les perturbations sur les diagrammes de diffraction. Quand les zones sont petites, I,(s) est dominee par I&) et on observe les pits de I,(s) sous forme de bandes laterales ou de raies de diffraction supplimentaires. Le nombre de raies de diffraction supplementaires augment lorsqu’augmente Qao et les bandes latirales ne sont qu’une &ape de ce processus. Quand la periodiciti du macroreseau est ptofondement perturbee ou lorsque Quo est grand, on observe direcrement I&) sous forme d’effets de diffraction. Cette thiorie explique raisonnab~ement la position dissyrn& trique des bandes laterales, fe dtplacement des raies de diffraction principales au tours du vi~iliissement et la transforma~on des bandes laterales en raies de diffraction de phases metastables au tours du vieillissement. Zusammenfassung-Modulierte Strukturen werden als ein Makrogitter betrachtet, in dem jeder Gitterpunkt ein “Einheitsgebiet einer Konzentrationsvariation” (abgekiirzt “Einheitsgebiet”) ist; der mittIerc Gitterparameter ist Qao Die von modulierten Strukturen reflektierte Riintgenstrahlintensitlt 1,(s) ist gegeben durch I,(s) = I&)*1,&). I,(s) ist die Streuintensitlt des “Einheitsgebietes”, I,,(s) ist diejenige des imaginlren Makrogitters, in welchem jeder Gitterpunkt der Einheitsstreufaktor ist. Als “Einheitspebiet” wird ein Gebiet mit einem Zonen-Komplex gewiihlt. Es wurden die Einfliisse des Gitterabstandes im Uakrogitter und dessen Stiirung auf die Beugungsdiagramme diskutiert. Sind die Zonen-Komplexe klein. wird i,(s) von I,(s) bestimmt; es werden Maxima von I,(s) als Seitenbgnder oder zusstziiche Beugungslinien beobachtet. Die Zahi zudtzlicher Beugungstinien nimmt mit zunehmendem Gitterparameter Oa, zu; typische Seitenblnder wird auf ein Zwischenstadium dieses Prozesses beobachtet. 1st die Per&dititat des Makrogitters stark gestort, oder ist Qao grol3. dann wird I,(s) direkt in der Beugung beobachtet. Die vorgelegte Theorie erkiPrt zuf~edenstellend die Asymmetrie der Seitenblnder in ihrer Position, die Bewegung der Hauptbeu~ungslinie mit der Alterung und den Wechsel der Seitenb%tder in Beugungslinien metastabiler Phasen wiihrend des Alterns.
I. INTRODUCTION
of mirror reflection across the main diffraction line to the low-angle side-band. The side-bands change The formation of modulated structures in aged into fixed diffraction lines during aging. The main difCu+!/,Ti and Ni-lO”/;;Ti alloys has been verified by fraction line moves to a high angle side with aging means of X-ray diffraction and electron microstime, after a certain stage was reached which is detercopy [l-15]. On the diffraction patterns of these mined by aging temperatures. The X-ray diffraction alloys the following are noted. The side-bands are theory of modulated structures must be able to broad and asymmetric in position, intensity and half- explain these phenomena on appropriate solute conwidth: the high-angle sideband is not in the relation centration variation models. 295 A..\,. 253-4
‘96
TSGJIMOTO,
HASHIMOTO
AND
In modulated structures a local concentration variation or a ‘unit region of concentration variation’ (abbreviated to a ‘unit region’) is repeated at semiregular intervals. The network of ‘unit regions’ can be regarded as a macrolattice. in which each lattice point is the ‘unit region’. In the present paper, the effects of the lattice spacing of the macrolattice and its disturbance on diffraction patterns have been discussed. A small region of solid solution containing one zone-complex has been chosen as the ‘unit region’ to explain the concept of the present theory and to depict the essential features of the diffraction patterns from modulated structures. Although the treatment is simple, this theory gives a general view of various aspects of diffraction effects by modulated structures and makes it possible to explain varieties of sidebands. Finally, diffraction patterns of aged Cu-4”,;Ti and Ni-lO%Ti alloys have been discussed. Up to this time various solid solution models with solute concentration variation have been proposed [16-281. However, most of them have not given any explanation for the asymmetry in position of sidebands. Some researchers explained this asymmetry by assuming a random distribution of zone-complexes E23.241. However, this concept does not agree with the observation of periodic microstructures in sidebands alloys. Also, behavior of side-bands at the later stage of aging have not been discussed in all previous papers. De Fontaine simulated diffraction patterns from modulated structures by using an electronic device [25]. The present paper gives a clear explanation based on an acceptable physical mode1 for some of his results. The germ for the concept of macrolattice can be seen in some papers [Z, 21,26,28-j. In this paper this conception is identified clearly and is used completely. Although the present theory is described, for simplicity, for one dimensional model except section 11.2.4, it can be easily extended to three-dimensional model. We have only to calculate in the same way the scattering intensity from a three-dimensional ‘unit region’ and a three-dimensional macrolattice, instead of one-dimensional treatment. II. THEORY 2.1 Fandamentals~or
di~~uctjon by adulated
structure
The amplitude of X-ray diffracted from a crystal lattice, A(s), is given by no-1 A(s) = C fn(s)-exp(-2xisr,), (1) n=O in which s = 2.sin B/E.,@is a diffraction angle, and j. is a characteristic wavelength of radiation. f”(s) is a scattering factor of an atom at the nth lattice point whose coordinate is r,. If the effect of the variation of scattering factors is markedly smaller than that of atomic spacings, the equation (1) may be replaced by A(s) = fe (s) f 2 exp( - 2nisrJ, n=O
(2)
SAITO:
X-RAY DIFFRACTION
fo(s) is an average scattering factor of atoms in an alloy. The present treatment for diffraction of the modulated structure is based on the conception of the macrolattice. The scattering amplitude from the ‘unit region’, A,(s). is calculated using the equation (1) or (2). Then the total amplitude of X-ray diffracted from the macrolattice .4,(s), may be written as x0-
A,(s) = A,(s)* y
1
expj-%isR,),
(3)
,G‘O
in which R,v is the coordinate of the IVth lattice point of the macrolattice. When the lattice parameter of the macrolattice is Quo, .s,,A,(s)
=
=
x
1
v =0
exp( - h’sR,)
sin asiV, Qae sin nsQn e
+exp {nisQa,(.Y,
- 1)).
Here, a0 is an average atomic spacing of the alloy. The totaf intensity of X-ray diffracted from the macrolattice, I{(s), is given by I,(s) = f”(S) x I,(s).
(4)
I,(s) = ‘4,(s)*Ads)*
(3
Here,
and Ids) = AAs). A&)* = (sin nsNoQaO,‘sin ;isQae)‘. (6) f&s) may be regarded as the intensity of X-ray diffracted from the imaginary macrolattice, in which each lattice point is not the ‘unit region’ but the unit scattering factor. I,(s) is a periodic function and gives sharp peaks at positions of s = p/Quo. p is an arbitrary integer. In the modulated structure, however. *unit regions’ are not always distributed with perfect or sharp periodicity as atoms in a usual crystal lattice. Or rather, they are distributed in an intermediate state between perfect order and perfect random. Here. we define I&) as intensity of X-ray diffracted from the imaginary macrolattice with diffuse periodicity. Although the exact function form of ids) is unknown. I&) is considered to give diffuse peaks at the same positions as f,(s). We suppose that peaks of I&) become broader with increasing disturbance of periodicity (see Appendix). By replacing I,(s) by I,(s). the expression for the X-ray intensity diffracted from the modulated structure, I,(s), is obtained : (7) I,(s) = I,(s) x rdH. Two features of the diffraction theory for the modulated structure are noted. One is diffuse periodicity in the arrangement of lattice points of the macrolattice. This feature brings about diffuse diffraction intensity. The other is that the scattering factor of lattice points of the macrolattice is strongly dependent
TSUJI?vfOTO. HASHMOTO
A4‘D
S‘AIT0:
X-RAY DIFFRACTION
797
upon scattering angles in a complicated manner. This is caused by the fact that ,4,(s) itself comes from diffraction of a large group of atoms. Z.1 Scarcering tion pcirrern
intensity from ‘unit region’ and difrac-
Examples of solute concentration variation models are shown in Figs. 1 and 2. Here, solute concentration variations of Fig. 1 are perfectly periodic and those of Fig. 1 are semi-periodic with an average spacing QL%. If a wavelength of sinusoidal concentration variation. Quo. is taken as the ‘unit region’ as in Fig. I(a). I,(s) and I,(s) will have peaks at the positions of 11~1~I: p, Qno. Thus. I,,,(S)and side-bands are symmetrical in positions about the main diffraction line. In the cases of Figs. 1 (b) and (c). any region containing a wayslength of concentration variation, Quo, may be chosen as the ‘unit region’. In such cases peaks of I,(s) are at the positions of h/a,, 2 piQao, but those of I,(s) are generally not. However, as peaks of IAs) are extremely sharp in comp~ison with those of I,(S), we get pcdks of I,(S) at positions of I,(s) after all. It must be noted that side-bands arising from any perfectly periodic concentration variations are symmetrical in positions about the main diffraction line and that they are as sharp as normal diffraction lines. Local concentration variations in Fig. 2 are related to the configuration of G.P. zone. (a) corresponds to a 3-phase model and (b) to a 2-phase model. When atomic spacings are a, in the range from &I; to n&b + 11~~7~ as in Fig. 2(b) and the average atomic scattering factor is fl. the scattering amplitude from this part. il$), is given by A2(s)
=
f? . s’n aStlJnl sin xsffI x exp(-ZKisnbnb).exp:nis~~(n~
- 1);.
If any parts of the ‘unit region’ do not have a close value of atomic spacings, the scattering wave from the part of atomic spacings LZ?will not interact with those from other parts. Then, IL(s) = A3(s)*AI(s)* = ,‘i *[sin nsrlzn,/sin ASPS)‘.Hence. I,(s) for Figs. Z(a)
-X
Fig. 1. Periodic solute concentration variation models.
Fig. 2. Local solute distribution and ‘unit region’ in semiperiodic concentration variation models.
and (b) may be summarized as follows: Each part gives the intensity f’ -(sin nsna;‘sin nsn)’ in which n and f are the number of atoms of the part having an atomic spacing a and the average atomic scattering factor of that part, and I,(s) is the sum total of the intensity diffracted from each part. f’ (sin ~snccjsin RSFZ& gives peaks of maximum intensity 6j2 and halfwidth I;nu at positions of s = q!u. 4 is an arbitrary integer. The peaks become stronger and sharper with an increase in n. ~i~raction patterns arising from the modulated structures having the -unit regions’ shown in Fig. 2(a) and (b) are shown in Figs. 3 and 1. respectively. For the calculation of I&s) equation (2) was used and Id(s) were considered to be those as indicated by chained lines. I,(s) are presented by broad lines or by hatched areas. It is noted that the main diffraction line or the high-angle side-band is weak or not observable for the concentration variation of Fig. 2(b). In the present theory ail kinds of local concentration variations may be adopted as a ‘unit region’. In the following explanation, however, the theory will be described for convenience on the basis of the 3-phase model of G.P.-zone in a semi-periodic array. This model can explain most reasonably the broadness and the asymmetry in position of side-bands, the existence and the movement of a main diffraction line with aging, and the change of side-bands into fixed diffraction lines. 2.3 Some specific aspects motiulnred structures
of diffraction
eficrs
from
Modutated structures are formed during aging aft.er quenching. We choose a region containing one zonrcomplex as the ‘unit region’. The size of the zonecomplexes and the distance between them increase with the progress of aging. We do not obtain enough information concerning the relationship between aging time and degree of periodicity in the distribution of zone-complexes. Ardell and Nicholson described that the array of ‘unit regions’ in a Ni-Al alloy is at random at the early stage of aging and becomes regular with aging time [303. On the other hand, the array will be relatively regular from the early stage of aging if an alloy decomposes spinodaily. This problem is related to the origin of alignment of decomposition products and decomposition mechanism.
-----+
Intensity
,pp
v, .;-+----.-_.>8p - ~-~_~+~_._.A--’ i !
\ \\ :I
-3.8 ‘1 +._.-\
:\ \ ---.-__
a-
-
943 s
I-
\
&I- I
+_
/
:i
I
-
rd‘
g
8
f&j g
F
a
: -*Intensity =i ___._.._..__________________ 0
St-
3
3-
1 -
‘\ ;
2 tl
‘4;_~___--_____-__‘__._ x __._ ___ __\_
:“-. . .._
I f_____ u
i
:\ : :, ! ~~,+-.-._._._._._q-
cn,
TSt’JIMOfO.
HASHI~OT~
.VGDSAITO:
X-RAY DIFFRACTI04‘
299
Fig. 4. X-ray intensity diffracted from clusters as in Fig. I(b) in a periodic array. a2 = 3.675 A, a; = 3.615 A and
Q = 138 are assumed. The effect of the degree of periodicity in the distribution of zone-complexes on diffraction patterns is shown in Fig. 3, in which (a) represents the diffraction pattern from zone-complexes in sharply periodic array and {b) and (c) represent those from zone-complexes in semi-periodic array. With increasing diffuseness of periodicity. each peak in 1&s)becomes broader and, as a consequence, the side-bands become broader and more asymmetric. In the case of a nearly random distribution, Cd),the foot of each peak of Id(s) falls on the foot of the nearest peak so that Id(s) becomes nearly constant. In this case, I,(s) can be determined directly from the diffraction pattern. The change in symmetry of side-bands corresponds to the change of the dominating factor on diffraction from f,(s) to Us). Figure 5 shows the effect of the distance Qao between the nearest zone-complexes on diffraction patterns. Here, 1&s) is assumed, for convenience, to have sharp peaks. Except the main diffraction line, one additional diffraction line is observed in (a), three lines in (b), and many lines in (c). It is clear that jj the number of additional diffraction lines increases 11 as Q increases and that typical side-bands, in which one additional diffraction line is observed on each iI j ! side of the main diffraction tine, are only a stage in i! the increasing process of the number of additiona 1 ! diffraction lines. In the case of a large Q value, (d), i! additional diffraction lines are so many and close that i1 they do not give virtually any effects on the diffraction i! pattern and the same diffraction effect as I,(s) is j! observed. i ! increase or decrease in solute concentration of each d zone of the zone-complex is reflected in the position of ‘peaks of I,(s). Increase in the size of the zonecomplex is reflected in the sharpness of peaks of I,(s). For small zone-complexes additional diffraction -s lines are observed in a wide range of Q values, i.e. Fig. 5. X-ray intensity diffracted from zone-complexes in from the early stage of aging to the later stage. For a periodic_array. The-same atomic spacings as in Fig. 3. large zone-compieaes additional diffraction lines are Q are 69 in (a). IiS in (b), 212 in fc) and 917 in (d).
300
TSUJIMOTO,
H~SHI~lOTO
AND
observed in a narrow range of Q values, only in the relatively larer stage of aging and they become strong rapidly during short time of aging. There will be various combinations of the size of zone-complexes. the distance between them and the degree of periodicity. If the periodicity in Fig. S(c) were diffuse, the feet of the peaks of I,(s) would overlap each other and I,(s) would become nearly constant. In such case I,(s) would be observed as the diffraction pattern. If the periodicity in Fig. j(b) were diffuse, the two peaks on the low-angle side would join together and a broad diffraction band would be observed. Although the asymmetry of the side-bands is only in intensity in Fig. 3(a), it is in position and in intensity in Figs. 3(b) and (c). This means that, for the side-bands showing the asymmetry in position, it is not appropriate to assume perfectly periodic array of zone-complexes. Figures 3 and 5 are drawn in the case of c1 - ca = 2 (q, - ct). In those circumstances the asymmetry of the side-bands increases with the increase in Q. This agrees with the experimental results described Iater. 2.4 ~l~re~t~ur~o~ dijj%action lines
between
side-bands
and
~~r~ul
The total intensity diffracted from the modulated structure is given by the equation (7). In extreme cases, the positions of diffraction lines are determined by the sharper peak of the two functions, Id(s) and I,(s). Usually, the dominating function changes gradually and continuously from I,(s) to I&) or vice versa. Thus. it is necessary to determine which function is dominant on a diffraction pattern. We define the patterns dominated by r&s) as side-bands and those dominated by i&) as normal diffraction lines. The reciprocal lattice points (abbreviated to relpoints) for the solid solution of the average atomic spacing a0 are indicated by the indices hkl. The reciprocal lattice of the macrolattice (abb. to sub-relattice) has the lattice spacing l/Quo and some of the subtelattice points (abb. to sub-relpoints) coincide with the hkl relpoints. Sub-relpoints around a relpoint are indexed as h’k’l’ by considering the hkl relpoint to be the origin of the sub-relattice. An hkl relpoint is observed as the main diffraction line at the scattering angle 8 on a powder X-ray diffraction pattern and an h’k’l’ sub-reIpo~t belonging to the hkl relpoint is observed as the diffraction intensity at the scattering angle 0 + AB1. Then, Q and AOl is related by
SAITO:
X-RAY DIFFRrtCTfON
a zone is a + Aa and that of a matrix is a, the diffraction line of the zone at the scattering angle 8 + A-\8a and that of the matrix at the angle 6 is related by AtJJtan 0 = Ann = constant.
When a region containing one zone-complex and matrix is chosen as the ‘unit region’ and the atomic spacings of inner- and outer-zone are n, and a,, the equation (9) holds for each zone of zone-complex. In the present model the reciprocal lattice of the ‘unit region’ consists of the relpoints for each zone of zone-complex and the matrix. Diffraction effects are observable only when a sub-relpoint coincides with the reciprocal lattice points of the *unit region’. All the additional diffraction lines except the main diffraction line are related to the relpoints of each zone of zone-complex. We can expect at the early stage of aging that the relpoints for the zone-complex are diffuse and cover some sub-relpoints and, as a consequence, the side-bands are frequently observed. The relpoints for the zone-complex become sharp at the later stage of aging. In this case we can observe the additional lines, only when the sub-relpoints are diffuse or dense. Some of the diffraction patterns have an intermediate character between side-bands and the diffraction lines of zone-complex. That is, the intensity peaks of I,(s) are between the peaks of I,(s) and those of f&s). The diffraction patterns according to the equations (8) and (9) are compared in Fig. 6. In the case of ci - co > co - ca, the peaks of I,(s) on the highangle side are closer to the main diffraction lines than those on the low-angle side. This brings about the asymmetry in the positions of the peaks of I,(s). A& becomes larger with the increase in Miller indices and this brings about the dependence of Q on Miller indices. Thus the values of Q estiniated from the sidebands of high indices have a tendency to be small. (200) side-bands are in a special situation. A62 of them, especially one on the high-angle side, are markedly smaller than ABi, and hence the estimated value for Q tends to be large. (113) side-bands observed usually correspond to h = 3, hence the estimated values of Q is not so small as anticipated in the case ofh=l.
(2201 (311)
which is reduced to the formula well known for sidebands
(222)
when h’ = 1 and k’, 1’ = 0. On the other hand, I,(s) is the function of the same type with the diffraction effect from ‘unit regions’ at a random distribution. When the atomic spacing of
I,1 II/ I l&l
,ll/!I
(111) 000~
Q = (hh’ + kk’ + ll’)*tan6/(na + kZ + 1’).A6, (8)
Q = h*tan 0/(h’ + k? -t 12)*A0,
(9)
1
9,
I
1
I ti
.-
-S Fig. 6. Comparison between side-bands or IAs) (solid Lines) and diffraction lines of zone-complexes or I,(s) (broken lines). The same atomic spacings as in Fig. 3 and Q = 40 are assumed. Id(s) is given for h’ = 1 and k’. 1’ = 0 except (113), for which h’ = 1 and k’, 1’ = 0 (far from M) and I’ = 1 and h’, k’ = 0 are given.
TSUJIMOTO,
HXSHIMOTO
III. EXPERIMESTAL RESULTS DISCUSSIOS
Ah-D
AXD
Diffraction patterns of the Cu-P~Ti alloy and the Xi-10aOTi alloy have been examined using a Cukr, radiation and a Guinier type camera. The side-bands of the Cu39;Ti alloy are shown in Fig. 7. The side-bands are weak. diffuse, relatively symmetric and in a large separation from the main diffraction line at the early stage of aging. They approach the main diffraction line with the progress of aging and become intense. sharp and asymmetric. The high-angle side-band is always intenser, sharper and nearer to the main diffraction line than the low-
SAITO:
X-RAY DIFFRACTIOI\;
301
angle side-band. At a certain stage the low-angle sideband broadens suddenly and then the side-bands are replaced by fixed diffraction lines. The main diffraction line moves to a high angle side from the beginning stage of a_tig at temperatures over 43C. but it begins to move at the later stage of aging at temperatures under 43’C. The main diffraction line keeps moving until it reaches the fixed diffraction line on the high angle side. Intensity of the main diffraction line decreases during aging and this phenomenon is remarkable in the process of the movement of the main diffraction line upon the low temperature aging. We consider that the zone-complexes aggregate during aging and that c2 - c,, > c,, - c2. The broadness of the side-bands represents that the zone-com-
Fig. 7. Side-bands of a Cu--l”/;Ti alloy
302
TSUJIMOTO. HASHIMOTO A>B SAITO:
plexes are in a semi-periodic distribution. The iixreasing asymmetry of the side-bands with aging is attributed to the effect of I,(s), whose peaks become sharper with the increase in the size of the zone-complexes. With the progress of aging, a situation similar to Fig. S(b) is realized. As the periodicity is not so sharp in the alloy, two diffraction lines on the low angle side in Fig. 5(b) are observed as a broad band With further aggregation the situation of Figs. 3(d) or S(d) is realized and I,(s) is directly observed. Thus, the fixed diffraction lines are those arising from the inner- and the outer-zone of the zone-complex and their positions correspond to the atomic spacings of each zone. As the inner- and the outer-zone are in a metastable equilibrium, they are named xl-phase and x,-phase, respectively. The movement of the main diffraction line to the high angle side represents the decrease of solute concentration of matrix. The coincidence of the main diffraction line with the a,-phase diffraction line means the achievement of a 2-phase structure.
X-RAY DIFFR,KITON
The sharpness of the side-bands and the weak asymmetry in their positions may indicate that the distribution of the zone-complexes in the Ni-loll/,-Ii alloy is more periodic than that in the Cu+Ti alloy. The fact that only the moving main diffraction line and the zz-phase line are observable in the course of aging may mean that the transfer from the 3-phase structure to the 2-phase structure takes place at the relatively early stage of aging in this alloy. Thus the patterns in such stage are considered to correspond to Fig. 4. IV. SU~IMARY
The side-bands observed for the Ni-lOF
The network of ‘unit regions’ is regarded as the macrolattice. in which each lattice point is the ‘unit region’ and the average lattice parameter is Que. The X-ray intensity diffracted from modulated structures, I,(s), is given by us) = f&)-l,(s). r=(s) is the scattering intensity from the ‘unit region’ and IAs) is that from the imaginary macrolattice, in which each lattice point is the unit scattering factor. I,(s) gives diffraction peaks at positions of p/a. where a is atomic spacing for each part of the ‘unit region’. I,,(s) gives diffraction peaks at q/Quo. p and q are arbitrary integers. The peaks of I,(s) are diffuse when the size of the part having the same atomic spacing is small. The peaks of Z,,(s)are diffuse when the periodicity of the macrolattice is disturbed. In the present paper a small region containing one zone-complex has been chosen as the ‘unit region+. Effects of the lattice parameter of the macrolattice, its disturbance and the size of the zone-complexes on diffraction patterns have been described. When the peaks of r,(s) are diffuse and those of I&) are sharp, the diffracted intensity I,&) is mainly dominated by I,(s). In this case, the peaks of I,(s) are observed as side-bands or additional diffraction lines. The number of additional diffraction lines increases with an increase in Qao and side-bands are observed at a stage of this process. When the periodicity of the macrolattice is perfect and the zonecomplexes are small, the side-bands have good symmetry in position about the main diffraction line. When the peaks of I&) are diffuse or Quo is large, I,(s) becomes constant and I,(s) is observed directly as the diffraction effects. The effect of the sharpness of the peaks of I,(s) or I,(s) and the effect of the Miller indices of the main diffraction line on diffraction angle of I,(s) have been discussed. The present theory can explain reasonably the following phenomena in the Cu-4% Ti and the Ni-loo/ Ti alloy: the asymmetry of side-bands in position. intensity and halfwidth; the movement of the main diffraction line to the high angle side with aging; the change of sidebands into diffraction lines of the metastable phases during aging.
Fig. 8. Diffraction pattern in the nei~hbourhood of the (200) main diffraction line of a Ni-LO?;Ti alloy plate aged at 600’C for 200 hr.
1. K. Saito, K. Iida and R. Watanabe, J. Japan Inst. Merds 31, 631 (1967) (in Japanese); Eans. Nat. Res. Inst. lwrtais 9, 267 (1%7).
3.2 D~~action
pntterns
of
Ni-lOT
REFERENCES
TSUJIMOTO.
HASHIMOTO
AND SAITO:
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9. T. Miyazaki, E. Yajima and H. Suga. Trans. Japan Inst. Metah 12. 119 (1971). 10. S. Nishikawa and K. Nagata Trans. Japan Inst. Mera/s 17, 277 (1976). 11. S. Ikeno, S. Saji and S. Hori, J. Japan Inst. Metals 38, 446 (1974) (in Japanese). 12. E. G. Nesterenko and K. V. Chuistov, Fiz. Merei. ,Vferuliooed. 9, 140 (1960). 13. H. Suga T. Miyazaki and E. Yajima, J. Japan Inst. .Clrtals 34, 1190 (1970) (in Japanese). I?. C. Buckle. B. Gentry and J. Manenc, Rec. Met. jl. 147 (1959). 15. D. H. Ben Israel and M. E. Fine, Acta ,Lfet. 11. 1051 (1963). 16. V. Daniel and H. Lipson. Proc. R. Sot. (A) 181. 367 (1943). 17. M. E. Hargreaves. .4cm Cryst. 301, 4 (1951). 18. D. Balli and M. Zakharova. Dokl. Akad. Nauk. USSR 96 453, 737 (195-t). 19. T. J. Tiedema. J. Bouman and W. G. Burgers, Acta Jlrt. 5. 310 (1957). 20. E. Biedermann, tlcta Cryst. 13. 650 (1960). 21. Y. Fukano, J. phys. Sot. Japan 16. 1195 (1961). __. ‘7 M Hillert, M. Cohen and B. L. Averbach, Acta Met. 9. 536 (1961).
X-R.AY DIFFRACTION
303
23. A. Guinier. ;Lcrcr.ifrr. 3. 510 (1955). 2-t. J. Xlancnc. C. r. 4cud. Sci. 248. 1Sll 11959). 25. D. de Fontaine. Locczl .4romic .4rrumqemenrs Sidied b) X-Ruy Dltfktion (edited by J. B. Cohen and J. E. Hilliard). pp. 51-9-t. Gordon & Breach. Sew York (1966). 26. Yu. D. Tiapkin and M. V. Jibuti. .4cra .\frr. 19, 365 (1971). 27. P. M. de Wolff. .4ctn Cr)sr. (A) 30. 777 (1974). 28. H. Bohm, rlcra CrFst. (A) 31, 621 (1975). 29. A. J. Ardell and R. 8. Nicholson, .-lcta ,Lfer. 11, 1295 11966). APPENDIX In order to exolain exoerimental facts. l.,(s) must be a periodic function’which is repeated by a small change of s and has diffuse peaks. From this point of view. we introduced the concept of the macrolattice and the diffuse periodicity as a convenient means. We may suppose the following model as an intermediate state between the perfect periodicity and the perfect randomness. That is. the actual center of the :vth ‘unit region’ is displaced by J. distance Ax,” from the Xth macrolattice point, the probability for large /A.u,~~is lower than that for small I Axu’. and 4”- L 7 A.u, = 0. VT” The displacement ratio to the lattice parametsr. l.x,v, @I,!. can be much larger than that in high temperature crystal. The broadness of the peaks from the macrolattice comes from the fact that in
the maximum value of each term except Ar, = 0 is not concentrated at positions s = p,Qa,. We assume Id(s) as chained lines in Fig. 3.