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On tri-axial periodic modulated structure in metallic alloys
ON TRI-AXIAL
PERIODIC
MODULATED
STRUCTURE
IN METALLIC
ALLOYS*
YU, D. TIAPKINT and M. V. JIBUTI: The calculation of the diffuse scattering of X-rays (t,he intensity of the satellite reflections about the Bragg reflections from supersaturated solid solution) has been carried out for the description of the structure of some metallic alloys in the initial stages of the decomposition of the supersaturated solid solution with the help of t,he model of a tri.axial modulat,ed periodic structure”*2’ (the modulation of the interplanar spacing and the scattering power in three (100) directions simultaneously). The comparison of the result,s of quantitative measurements of the satellite intensity in various chrections away from the Bragg maxima (in the dire&ions (100) and (110)) for the initial st,age of aging of the F+Be alloys with 16.5 and 19.5 at. y/, Be after annealing at, 400” with t,he results of calcuiat~ion has proved t,he correctness of the proposed model. The study of the satellite intensity in various dire&ions depending on different, parameters of the model (on the period of modulation L and the degree of modulation ha/a) permits an explanat,ion based on t~heproposed model, the observations in t,he previous1.x studied alloys with the modulated periodic strucbure (in such alloys as “CuNiFe,” “CuNiCo,” “ticonal ot)hers), in contrast, to the Fe-Be alloys, of satellites only in the din&ions (100). STRUCTURE
MODULEE
TRIASIALE PERIODIQUE METALLIQUES
J)ANS
LES
ALLIAGES
Le calcui de la diffusion anelastique des rayons X (l’intensite des reflex-ions sat,ellit,esaux environs des reflexions de Bragg B partir de la solution sursatu&e) a PtP men& it birn dans Ie but de deerire la &rueture de certains ailiages m~talliques dans les tout premiers stades de decomposition de la solution solide sursatureehl’aidedu modele d’unestruct,ure periodique modufee trialiale”*2’ (modulation de i’espacement interplanaire et de la puissance de diffraction clans trois directions (100) simultanees). La comparaison des resultats des mesures quantitatives de l’intensite satellite dans des directions variees loin des maxima de Bragg (dans 10sdirections (100) et (110)) pour les stades initiaux du vieillissement des alliages Fe-Be avec 16,5 et 19,50/:,at. Be, apres recuit a 400”, avec les resultats dea calculs a montri? quo le modele propose est correct,. L’etude de l’intensite satellit,e dans des directions variees dependant den differents parametres du modele (la periode de modulation & et le degre de modulation Acs/a) permet une interpretation des satellites seulement dans les directions (loo), basee sur le mod& propose et les observations dans les alliages etudies amerieurement aver la structure periodique modulee (dans des alliages tels que IR “CuNiFe.” le “CuNiCo,” le t~iconal” et, d’autres), en contraste avec les a&ages Fe-Be. EINE
DREIACHSIGE,
PERIODISCHE,
MODULIERTE LEGIERUNGEN
STRCKTUR
IN
~~ETA~LISCHEN
Die Berechnung der diffusen Rontgenstreuung (Intensitat der Satellitenreflexe der Bragg-Reflex8 einer iibersattigtan fasten Losung) wurde zur Beschreibung der Struktur einiger metallischer Legierungen in den Anfangsstadien der Entmischung der tibersattigten Liisung mit Hilfe des Modells der dreiaachsigen, modulierten, period&hen Struktur (i.2) durchgeftihrt (Modulation des Netzebenenabstandes und der Streuamplitude in den drei (loo)-Richtungen gleichzeitig). Ein Vargleich der quantitativan Messungen der Satellitenintensit(iten in verschiedenen Richtungen im reziproken Gitter (van den Bragg-Maxima in Riehtung (100) und (110)) fiir die Anfangsstadien der Auslagerung van Fe-Be-Legierungen mit 16,5 und 195 At. 0/0Be (nach Anlassen bei 409’) mit den berechneten We&en hat gezeigt, datl das vorgeschlagene Model1 richtig ist. Die Untersuchung der Satellit~enintensit~t in verschiedenen Richtungen und als Funktion der verschiedenen Parameter des Modells (~odulat,ionsperiode A, Modulationsgrad A+) erlaubt eine Erklarung der an friiher untersuchten Legierungen mit modulierten periodischen Strukturen (wie “CuNiFe,” “CuNiCo”, “ticanol” und andere) gemachten Beobachtungen, bei denen, im Gegensat)z zu Fe-Be-Legierungen, nur in (100).Richtungen Sat~elliten auftreten.
As was pointed out in earlier papers(102)the model of the uni-axial modulated (along directions (100)) structure which is usually proposed for the expIanation of X-ray data (side-bands near main reflections from supersaturated solid solutions) taken inthe early stages of aging of some alloys (such as “CuNiFe”, “CuNiCo”, “ticonal” and many others) in many cases contradicts the experimental data (both electron-microscopic and X-ray). For a more correct explanation of the available experimental data a new model assuming simultaneous modulation of the in~rplanar spacing and scattering power in three directions (100) (periodic or not periodic depending on an alloy’1~2~4))has been proposed. In present paper we give the results of the calculation of * Received January 5, 1970; revised August 25, 1970. t Moscow, Radio str. 23, Central Scientific Research Institute for Ferrous Metallurgy (Tsniitchermet). $ Georgian PO&technic Institute. ACTA
METALL~JR~ICA,
VOL.
19, APRIL
1971
reflection intensityof this model for the case of periodic modulation in aging Fe-Be alloys and in some other alloys, too. The results of the calculat.ion are compared uith tSheexperimental data. Our model that assumes the modulation of interplanar spacing and scattering power in three directions (100) simultaneously, for the copper, nickel or iron base alloys that are known to have anisotropy of elastic constants of C,,C,, c 2C,, type agrees with the available thermodynamic theories of decomposition of s~lpersaturated solid solutions (see for examplef5)). But our calculat,ion will be done not for the sinusoidal waves but for the waves in the (100) directions, which are rect’angular, i.e. the constancy of the (100) interplanar spacing at the definite parts along (100) is supposed. In this respect our model is similar to the Hargreaves scheme,cQ but we suppose the modulation of the interplanar spacing along three satellite
365
In our case the function
U, will be a periodic
funa-
tion with the period along the (100) axes being equal
L = (2M -t l)a = Q . ti. The structure factor for such a unit cell (the size of the complex is equal 2M&. see Fig. 1) can be expressed as : 2 F(J) exp (-&r&r’)
P,(S) =
~‘comp1ex = ;
F ;
I1p t
hF(%
Y>z)l
x exp [--~2n+&,
+
u,,,)l
x exp 1.--2~~~&
+
Uk.Jl
x
+ Un-,*)]
exp [-27ru,(Z,
where s,, slit s, are the components
FIG. 1. Model of tri-axial periodic modulated structure [only half of one “unit crll” of the periodic structure
let us neglect AF = 0).
divided by plane (010) is given].
data. (see Fig. I). This model may
also arise from consideration inkfaces.
But as will be shown
tangular modulation stages
of
in Fe-Be
decomposition
reflections
of
precipitation model
(or
be observed.
the interplanar
below,
of phase such
rec-
alloys appears at early
long
equilibrium may
spacing
before even
the
X-ray
intermediate)
According
to this
aIong each of these
changes so that at some part (LV -
2~1 of interplanar following
of late &ages of spinodal
as the result of sharpening
decomposition(a)
lattice,
pressed in units of ii*, F(J) is the structure the k unit, cell.
directions simultaneously
of the s vector
along t,he axes x, y3 z of t,he reciprocal
Oi.
/ IO101
spaces)
it is equal
part (2m of interplanar
For the simplicity the modulation it follows
cliagram for the Fe-Be composit,ion
of scattering
with experimental
at, 400°C t,wo phases arise. which
solution-nearly
“CuNiFe’
and “CuKiCo”
The
easy
= &4 per cent. For 450°C AFlp
we have still lower value: are nearly
differ
&-phase,
nearly 23 at. %
15 at’.% Be.
calculat’ion gives us AF/P values
states
al10ys,(~) as a result of the de-
ordered after the FeAl type, contains
These
of
power
from the metastable
from one another in content. of Be slightly: Be, a-solid
ex-
factor
of the calculation
This is in accordance
Indeed,
(1)
=
53
per
cent,.
equal to the values for the
alloys.
See also.(2)
For the whole crystal t,he scattering
a~ilplitude is:
t,o nl, at t,he
spaces) it. is equal to
repeated along the (100) (L2. This is periodically directions with the period of (2M + 1)s = L, where d is an average (100)interplanar lattice period).
The complex
spacing
(the average
shown in Fig. 1 is one
The equation for intensity
is :
repeating unit of such a periodic structure. The law of the change of atom coordiliates periodic structure
is given in the following
in tohis
equation: sin2 ns,QNz
X
Fe is a vector k-point
which characterizes
the position
of
of the mean lat,tice, U, is the atom displsce-
ment from k-point of the mean lattice; are the components
of this displacement
x, y, Z, Fk and U, are measured
u&,~.up,*, uk.% along the axes
in units of 8.
In
accordance with our supposition of the modulation of the interplanar spacing along three (100) axes simultaneously, components of ukuk,%, uk,?, and uk,, depend only on 2, y and z correspondingly, i.e.
sin2 nsZQN,
_._... .
sin2 T$~Q
sin2 TS,&
(2)
where N,. N, and N, are the numbers of periods of the modulakd
structure
along the axes x. y. z.
equation (2) follows that bhe value I(S) maxima
in the points of reciprocal
From
= /A( s)i2 has
space, for which
csx= h -1: (n/Q),where n is a whole number characterizing the order of satellites, and h is an order of reflection of the main maximum. It follows from equation
(2) the tri-axial
modulated
cause
lattice network
structure
space
the
must
appearance
of
in the the
periodic reciprocal
complicated
of satellite reflect.ions with coordinates
not,
TIAPKIS
AND
JIJ31’TI:
THI-AXIAL
t’ERI0DI.C
only {+*8JQ1 0: 0) as is usually observed, j&m,/&,
I&/&>
O> and
For the determination
modulat,ed
!I%/&).
of the intensity of these satellite the value of F,
with the help of equation
(1).
For this purpose the
of the sums of the following
type
structure
IS
must
be done:
observed
of interplanar
the whole complex
by 2M, cz is a mean lattice param-
eter, a, and a2 are spaces between
the planes
{loo]
outside and inside of the central part of t,he complex. As is seen from t*he model constant part
(Fig. l), these spaces are
along the direction
of the
summation
complex
(lOO> inside the central
(nz) and outside
(n,).
Upon
we get:
reflections
are
At this later stage we
rather diffuse reflections
which correspond
onal (with period
aT1 = a1 and cT1 = oa and aT2 =
a2 and cT2 = n,) conjugated of these
regions.
observed
A model
pictures
posed (the one tohat fully coincides Following, (*) let, us denote the number
stage with the
satellite
to two cubic (with periods a, and a,) and two tetrag-
interpretation
spaces in the central part of the complex by Xm, and in
367
XLLOYS
the so-called
where
seen has been studied in detail.
reflections it is necessary to calculate calculation
STRUCTURE
later stage t,hat follows
but
(I!I?~,,/Q, jl~z,/Q.
MOI)‘I’LATED
for
was pro-
with the model in
Fig. 1). As is seen from equations of the satellite reflections know the following
(3), (4) for the calculation intensity
parameters:
it is necessary
to
Q = 2M, 2m, 6, a,
and a2. Value Q = 2M (the period of the modulated struct’ure) is determined t,o the distances
from X-ray
data (according
of t,he satellite reflections
from the
main refle~t,ions) and from elect,ron-microscopic
data.
We note at once that these data are in satisfactory agreement.
The size of the complex
central part 2m
for Fe---Be alloys is simply estimated from the broadening of superlattice metastable
reflections
from the nuclei of the
phase with the C&l structure
which pre-
cipitate
in these alloys at the early stage.
position
of these superlattice
be determined.
reflections
The value ii. is determined
positSion of the main reflections the whole alloy) lattice. For the value of the scattering which
reflections A(s)
correspond
amplitude
to the maxima
we shall have the equation:
parameters
of our model
parameters
determined
of two compositions;
which
(4)
by equations such as
.y, = h + (n/Q,. We now use t)he above given equations of t-he satellit,e reflections
the &age of t,wo cubic and two tetragonal
up to the parameters t’he later stage-the
intensit,y for those correspond
to the alloys
alloy No. 1 with 19.5 at ‘A Be
are adequate
stages with intermediate we suppose
In this work the
scatt,ering
(the
increasing
of
30 2 30 2
1
1 1
2 -.-
min hr min hr
are given.
2.809 2.809 2.817 2.817
A
parameters
changes from the beginning
period at different~ stages of in t#he
phases which This change
(x1 and CL%.Thus
t,hat the whole process of the structural
these alloys was studied in Ref. 2.
aging and t,he manner of satellite disposition
to
reflec-
occurs so quickly that we do not usually observe the
ray diagrams
space of the matrix
aI and uZ tha,t correspond stage where independent
to our model are observed.
The structure of
lattice
phases.c2J
The same phenomeI]on takes place for value a2, i.e. the
and alloy No. 2 with 16.5 at. ok Be.
reciprocal
The latter
value is close to the value derived for the later titage-
tions from two cubic and two tetragonal for the cal-
while stTtdying Fe-Be
values of the modulation
(for
From the values & a2, 2N and
t,he d~eolnposit,ioxl at the initial stages quickly proceeds
= IE’I N1N,1L’,ct;,(s,)~,(~~)~~(s,)
where sZ, s, and s, are determined
culation
at the
of sat’ellite
from the
of the average
2n/, a value of a, can be easily calculated. points
From the
value aa ean
of the appearance
on ,X-
of the rather intense effects of diffuse satellite the
reflections)
modulat,ion
period
(and the value 2nc correspondingly) changes of parameters aI and a,.
2.76
140
2.76
200
2.76 2.76
80 1%
consists
in the
Q = 2%’ + 1
without significant The justice
of this
48 G 44 -____
x 12
ACTA
36X
METALLURGICA,
TABLE 2. The parameters
VOL.
of the calculated
19,
19’71
models of t,he modulated
structure 2m
Model
B C Z) ti: F
-
2.837 2.837 2.837 2.837 2.816 2.823
2.809 2.809 2.817 2.817 2.809 2.809
A
supposition intensities
is proved
2.7623 2.760 2.7670 2.7637 2.7966 2.7842
0.02659 0.02741 0.02485 0.02602 0.00691 0.01381
also by the calculation
of the satellite
reflections.
after
No.
(After
quality (alloy No. 2). In these cases the estimation of a the 10 step scale (10 steps are the intensity
1 and No.
values
2 (compare
of 2M,
treatments
obtained
at 400” for 30 min and
for 2 hr.
calculated
the effects
are revealed
in the form
most
clearly.)
2m, ii and a2 determined
experimentally
for these
(see Ref. 2) are given in
Table 1. The parameters of the calculated of different
the intensity
models are
satellite reflections
of the corresponding
nates are given with respect
main maximum) coordi-
to the position
of the
main maximum.
The results intensity
of the
(referred to
are given in Table 3. The satellite reflections corresponding
of The
given in Table 2. The results of the calculation intensity
Ss is seen from Table 4 satisfactory
1 and 2)
reflections
of the relative
of satellites (with respect to the correspond-
ing main maximum)
between
the
results.
measurements qualitative
are given in Table 4.
For alloy
of intensity
taken into account).
In other
more or less reliable quantitative microphotometer-either
were made with a
cases we could
not do
measurements
because
with a
of the nearness of
satellites to the main reflection and their incapability to be revealed
on
microphotograms
annealing at 400” for 2 hr, reflection of the absence of microcrystals TABLE
3. The relative intensity Reflection
[alloy
No.
and experimental table.
(IlO)] or because
of a sufficiently
high
of such measure-
agreement between theoretical
for the treatment
of alloy No. 2
of the intensity
X-ray
diagrams
However
that
the
it is easy to see in
intensity
of
satellites
(l/Q, l/Q, 0) is a little less than the intensity lites (l/Q, i/Q, 0) [or (i/Q, l/&,0)];
Let us here estimate contribution
The same occurs for alloy the quant,it,y of the possible of the scattering power
in the diffuse scattering intensity, above.
which we neglected
We shall calculate the intensity of the satellite [*
l/Q, 0, 0] of the reflection
model A (Table 2), providing cent (see above). equation
of satel-
this agreeswiththe
at 400” for 30 min.
of the modulation
that AP/P
In accordance
(1) F(z, Y, 2) = F,(z)
Reflection
200 for the = 14
+ F,(Y) + F&L (3), (4)
200, satellite coordinates
($&O)
(&OPO)
(&&O)
(&&O)
A B C ;
20 58.9 4.6 19.7 2.5
13 32.2 1.7 8.5 1.08
4.2 34.7 0.21 0.063 3.8
1.7 10.3 0.03 Kll
2.7 b:k
P
10.5
6.95
1.1
0.48
0.72
0.027 1.6
($o,o) 15 1890
(;,o,o) 165 4020
per
with our model in
for different models by equations
110, satellite coordinates
(;&O)
of
satellite (l/Q, l/Q, 0) and (l/Q, i/Q, 0) the value of less
(0,&O)
Model
only agree-
data is better than this is given in the
For example,
of satellites Za/Zhkl in per cent calculated
($o,o) (o,;,o)
not
quantitative
at 400” for 2 hr for the values
reflexions 1,
possible
the
exact
For the cases of visual estimation
the received qualitative
results of the calculation.
200 these measurements
were
and
where
ments and a large number of possible errors must be
No. 2 after the treatment
microphotometer.
data
cases
ment was achieved (the complication
110 and 200 for the treatment reflection
those
but also satisfactory
No. 1 after annealing at 400” for 30 min for reflections at 400” for 2 hr for
agreement was
experimental
For
than 1 step is given.
of the measurements
of the
tnain reflection).
Tables
these treatments
satellite
37.6 36.1 28.6 27.3 36.1 31.1
relative intensity was made visually and expressed by
the models for which the parameters
annealing
18 26 8 12 26 26
of the
are nearly the same as the parameters of structures of alloys
48 72 28 44 72 72
These results
will be given later. We calculated
138 205 82 127 205 205
($,o,o) 75 1020
(;,o,o) 44 1520
P,,
TABLE
_.-_
4. The comparison of the relative intensity of the satellite reflections I,/1 hkl in per cent or in 10 step scale* measured for alloys No. 1 and No. 2 and calculated in accordance with models A, B, C and D -._Reflection 200, satellite coordinates Reflection 110, satellite coordinates .._P -. -
(:3’0’0)(o,$o) (o,~~O)($o:oj ($;?o) (&go)
Tr~$y Alloy No. 1 400” 30 min Model A Alloy h’o. 1 400” 2 hr Model B Alloy x0. 2 4ooy 30 min Model C Alloy No. 2 4o0° 2 hr Model D
25
18
8
6
3
20 5b
13 4b
4.2 Pb
1.7 2b
2.7 26
58.9 tlb
32.2 SCIb
34.7 0
4.6
1.7
0.22
46 19.7
2b 8.5
lb 3.8
* At visual estimation
10.3 0 0.03
<
= (P -b- 3AF,fq,‘yy’yz’
+ (p + 3AF,)
x pl,$”. c&” . q&” --I-(E + 2AJ’, + AF,) x (q&l * P)vf* 47211 + f?Ji * 9;
* fPzf
+ qz’ . p; . rp,‘) + (P + AF, + zAF,t x (91,’ * 9y” ’ ye” + q&” * f&/’ * vz” + 9%” * hU . 9%‘) In our case (AF/P
1Q 0 0.08
i lb 1.6
($o,o) ($,o,o) (&o,o)
IQ
so
63
31
15 2000
165 3300
15 1300
44 1000
1890 -
4020 -
1020 -
1520 .-
-
-
I-_
-
-
-
_-
-
10 steps (lob) me the intensity of the main maximum.
F,, Fa depending only on x, y and z, respectively. In the center of the complex Fl(x) = F, = con&, outside of the center of the complex F,(x) = F, = con& The same is true for F, and F,. Upon summation in equation (l), taking into account (3), we get: F,(s)
(&!$‘O) ($‘&O) &o,o)
0%
= 1-4 per cent) AF,/P
= -AF,,J
E = 0.0133. Using the parameters of the model A (Table 2) we get for relative intensit,y of satellites (i/Q, 0, 0) and (l/Q, 0, 0) (in per cent of main reflexion) the quantities 169 and 74.5, respectively. In the approximation AF = 0 we get the quantities 165 and 75, that differ from the latter quantities not more than by 2 per cent. This is higher than the error of measurements of X-ray reflection intensity. Thus our approximation AF = 0 is correct enough. Thus, t*his study proves the correctness of the proposed model of the modulated structure for the case of the Fe-Be alloys. But formerly this model”) was earlier proposed for such alloys as Yiconal” and “CuNiFe,” for which the observed picture of diffuse
140 130 120 ,\" II0 1 IOQ-
h go? eo-Y 70 6050-
202224
9x
262830
10-3
3234
L,
Fra. 2. The change of the relative intensity of the various types satellite reflections of reflection (a) L, and (b) Au/&
8
Si
(110) depending on
scattering differs from that, for the Fe-Be alloys (for alloys of “ticonal” and “CuNiFe” types t,he socalled “cross” satellites, i.e. satellites with coordinates (l/Q, l/Q, O), (l/Q, l/Q, l/Q) and others, are not observed). The calculat,ions reported in the present paper allow us to explain this seeming disagreement. Let, us study within our model the change of the relat8iveint,ensity of satellites depending on the magnitude of the modulatio1~ period L = Q - 6 and the degree of modulation Aa@ = (a, - a&Z. In Fig. 2(a) the curves of dependence of the relative intensity of different satellite reflections of reflection (110) on the magnitude of the modulation period with given ratio 2Ml2m -_- 2.77 and An/a = 0.0267 are shown. It is clear that w&h small magnitude of modulat,ion period L only satellites in directions (100) from the main maximum, i.e. (l/Q, 0, 0) type. have noticeable inter&y. By increasing the modulation period, the intensity of bicross” satellit’es increases. When the magnit#udeof the modulation period is large enough (1; > 200 a, Fig. 2) some satellites [e.g. (l/Q, l/Q, O)] can become more intensive than satellites (l/Q, 0: 0), i.e. than those ones which are in direction {lOO>, and even more intensive than t,he Bragg maximum. This regularity is confirmed (as is seen from Table 4) by the studied Fe-Be alloys. These curves have the same character at other paramet’ers 2M/2n7 and Aa/& Thus, at small magnitudes of t!he modulation period, the intensit.y of “cross” satellites is so small that it is practically impossible t,o reveal them. The second important factor which changes the intensity ratio among various satellites is the degree of the modulation parameter Aa/& The dependence of the relative intensity of various types of satellites on this parameter is given in Fig. 2(b), where 2M/2m and L are constant parameters. They are equal t’o 2.77, 200 8, respectively. As is seen from Fig. 2(b), by increasing AajZ the intensity of the ‘icrosst’ sat,ellites increases faster than the “main” satellites (the satellites in the directions {loo)), so that with Aa/d = 0.0320 their intensity is comparable to the satellite intensity in directions (100) and with Au/E = 0.0330 the intensity of some “cross” satellites exceeds the intensity of the “main” ones. These curves explain why “cross” satellites may not be observed in such alloys as “ticonal”, “CuNiFe” and in other alloys with a modulated periodic structure. It is known that for these alloys the value of Aa/% does not exceed 0.010-0.015 and ratio 2M/2m usually varies from 2 to 4, i.e. near to value 2.7, used for the calculation of the curves given in Fig. 2(b). As is easily seen from this figure the intensity of the LLcross”
satellites relative to t~he intensity of the “main” satellites (satellites in the directions !lOOj); with values Aa/& in the given range of AajE values and even with such a modulation period as L = 200 A. will not’ exceed 2 or 3 per cent. It is necessary to conclude about’ t,he connection of the initial stage of decomposition of Fe-Be alloys (the so-called &age of modulated structure) with the stage of itlt,ermediate phases. As is seen from our investigations these stages differ chiefly only in thrb magnitndc of the modulaGon period. Indeed, with a sufficiently large value of &. t,he int’ensity of satellites increases so [see for example, Fig. 2(a)] that, it becomes much larger than t,he intensity of the main Bragg maximum. Wit,h further increase of the modulation period, i.e. with t,he &crease of the regions that compose the modulated stru(~t,~lre (two cubic and two tetragonal. see Fig. 1) the picture of satellites changes gradually into the picture of independent reflections from these regions. For the stage of modulated structure such independent’ reflect,ions can be seen also on the reflections w&h valnt~ XX2 that. are large enough.(2) CONCLUSION
The calculation of t,he diffuse scattering of X-rays with the help of the proposed model of tri-axial modulated periodic structure with periodic modulation of interplanar spacing and t,he scattering power along three directions {l@O) simult.aneously was carried out. The calculation showed that within t,he reciprocal lattice of such a cryst.al near the main (Bragg) maxima a complex spatial net of additiona, satellite reflections must arise, i.e. the satellite reflections removed from the main maxima not only in directions (100) but also in directions (llO), (111’: and others must appear. The quantitative measurement*s of t.he intensity of the satellite reflections for Fe-Be alloys with 16.5 and 19.5 at.% Be after annealing at 400” confirm the results of the calculation, proving t,he correctness of the chosen model. The calculation of the intensity of the various satellite reflections as a function of such parameters of the model as the modulation period L and the modulation degree Aa~~, allowed us to establish t,hat by increasing L and Aa~~ the intensity of t,he so called “cross” satellites (i.e. satellites in the directions (1 lo), (111) and others) at first is small in comparison with the intensity of the “main” satellites (in the directions (loo)), then it increases very fast so that under the definite values of L and Aa/d it may even exceed the intensity of the “‘main” satellites. These
TI.~I’KIS
ASD
sJI.Hl"L'I: THI-ASIAI,
PERIODIC
M01)L‘LA’l’El~
with modulated periodic structure (such as “CuNiFe”. “ticonal’” and others) the intensity of t,he “CuNiCo”. b‘cross” satellit>es must be negligible in comparison with t#he “main” satellites. Thus, during the experi-
ments rev~alcd
only
sat,ellit,es in directions
and tha,t is what, is ohserwd.
1100) should
bc
IS
ALLOYS
371
REFERENCES
cal~~llat,ions show &at, wit,h values L and &z/6 t,hat. correspond to the cases of the alloys studied previously
STRCCTlTRE
1. Kf. 3. %iIIK&Iil.11. 1‘. Em?
~~)Oltl~H~OBa --;SJtIicc:cP
178,
{lOcii).
2. AM.N.
,&KHC,YTA, K). ;J. Tmtmi- -tipmcTannorpa@wi, 13, 307 (1968). 3. V. DANIEL and H. LIPSON, Pm. R. Sot.18lA, 368 (1943). 4. HI. +!I,. THIIKHH. M. II. I’eopr~ee. M. R. ~Y$R@vTI~ GAlI CCCP 188, 80 (19cis).
5. J. W. CAHB, 24cta Met. 10, 179 (1962). 6. M. E. HARGREAVES. Aeta crystallogr. 4, 301 (1951). 7. M. \'.HEIMEXDAKL 11963i.
tend U. HET.BNER, Acta &let.11, 1119
PERIODIC
MODULATED
STRUCTURE
IN METALLIC
ALLOYS*
YU, D. TIAPKINT and M. V. JIBUTI: The calculation of the diffuse scattering of X-rays (t,he intensity of the satellite reflections about the Bragg reflections from supersaturated solid solution) has been carried out for the description of the structure of some metallic alloys in the initial stages of the decomposition of the supersaturated solid solution with the help of t,he model of a tri.axial modulat,ed periodic structure”*2’ (the modulation of the interplanar spacing and the scattering power in three (100) directions simultaneously). The comparison of the result,s of quantitative measurements of the satellite intensity in various chrections away from the Bragg maxima (in the dire&ions (100) and (110)) for the initial st,age of aging of the F+Be alloys with 16.5 and 19.5 at. y/, Be after annealing at, 400” with t,he results of calcuiat~ion has proved t,he correctness of the proposed model. The study of the satellite intensity in various dire&ions depending on different, parameters of the model (on the period of modulation L and the degree of modulation ha/a) permits an explanat,ion based on t~heproposed model, the observations in t,he previous1.x studied alloys with the modulated periodic strucbure (in such alloys as “CuNiFe,” “CuNiCo,” “ticonal ot)hers), in contrast, to the Fe-Be alloys, of satellites only in the din&ions (100). STRUCTURE
MODULEE
TRIASIALE PERIODIQUE METALLIQUES
J)ANS
LES
ALLIAGES
Le calcui de la diffusion anelastique des rayons X (l’intensite des reflex-ions sat,ellit,esaux environs des reflexions de Bragg B partir de la solution sursatu&e) a PtP men& it birn dans Ie but de deerire la &rueture de certains ailiages m~talliques dans les tout premiers stades de decomposition de la solution solide sursatureehl’aidedu modele d’unestruct,ure periodique modufee trialiale”*2’ (modulation de i’espacement interplanaire et de la puissance de diffraction clans trois directions (100) simultanees). La comparaison des resultats des mesures quantitatives de l’intensite satellite dans des directions variees loin des maxima de Bragg (dans 10sdirections (100) et (110)) pour les stades initiaux du vieillissement des alliages Fe-Be avec 16,5 et 19,50/:,at. Be, apres recuit a 400”, avec les resultats dea calculs a montri? quo le modele propose est correct,. L’etude de l’intensite satellit,e dans des directions variees dependant den differents parametres du modele (la periode de modulation & et le degre de modulation Acs/a) permet une interpretation des satellites seulement dans les directions (loo), basee sur le mod& propose et les observations dans les alliages etudies amerieurement aver la structure periodique modulee (dans des alliages tels que IR “CuNiFe.” le “CuNiCo,” le t~iconal” et, d’autres), en contraste avec les a&ages Fe-Be. EINE
DREIACHSIGE,
PERIODISCHE,
MODULIERTE LEGIERUNGEN
STRCKTUR
IN
~~ETA~LISCHEN
Die Berechnung der diffusen Rontgenstreuung (Intensitat der Satellitenreflexe der Bragg-Reflex8 einer iibersattigtan fasten Losung) wurde zur Beschreibung der Struktur einiger metallischer Legierungen in den Anfangsstadien der Entmischung der tibersattigten Liisung mit Hilfe des Modells der dreiaachsigen, modulierten, period&hen Struktur (i.2) durchgeftihrt (Modulation des Netzebenenabstandes und der Streuamplitude in den drei (loo)-Richtungen gleichzeitig). Ein Vargleich der quantitativan Messungen der Satellitenintensit(iten in verschiedenen Richtungen im reziproken Gitter (van den Bragg-Maxima in Riehtung (100) und (110)) fiir die Anfangsstadien der Auslagerung van Fe-Be-Legierungen mit 16,5 und 195 At. 0/0Be (nach Anlassen bei 409’) mit den berechneten We&en hat gezeigt, datl das vorgeschlagene Model1 richtig ist. Die Untersuchung der Satellit~enintensit~t in verschiedenen Richtungen und als Funktion der verschiedenen Parameter des Modells (~odulat,ionsperiode A, Modulationsgrad A+) erlaubt eine Erklarung der an friiher untersuchten Legierungen mit modulierten periodischen Strukturen (wie “CuNiFe,” “CuNiCo”, “ticanol” und andere) gemachten Beobachtungen, bei denen, im Gegensat)z zu Fe-Be-Legierungen, nur in (100).Richtungen Sat~elliten auftreten.
As was pointed out in earlier papers(102)the model of the uni-axial modulated (along directions (100)) structure which is usually proposed for the expIanation of X-ray data (side-bands near main reflections from supersaturated solid solutions) taken inthe early stages of aging of some alloys (such as “CuNiFe”, “CuNiCo”, “ticonal” and many others) in many cases contradicts the experimental data (both electron-microscopic and X-ray). For a more correct explanation of the available experimental data a new model assuming simultaneous modulation of the in~rplanar spacing and scattering power in three directions (100) (periodic or not periodic depending on an alloy’1~2~4))has been proposed. In present paper we give the results of the calculation of * Received January 5, 1970; revised August 25, 1970. t Moscow, Radio str. 23, Central Scientific Research Institute for Ferrous Metallurgy (Tsniitchermet). $ Georgian PO&technic Institute. ACTA
METALL~JR~ICA,
VOL.
19, APRIL
1971
reflection intensityof this model for the case of periodic modulation in aging Fe-Be alloys and in some other alloys, too. The results of the calculat.ion are compared uith tSheexperimental data. Our model that assumes the modulation of interplanar spacing and scattering power in three directions (100) simultaneously, for the copper, nickel or iron base alloys that are known to have anisotropy of elastic constants of C,,C,, c 2C,, type agrees with the available thermodynamic theories of decomposition of s~lpersaturated solid solutions (see for examplef5)). But our calculat,ion will be done not for the sinusoidal waves but for the waves in the (100) directions, which are rect’angular, i.e. the constancy of the (100) interplanar spacing at the definite parts along (100) is supposed. In this respect our model is similar to the Hargreaves scheme,cQ but we suppose the modulation of the interplanar spacing along three satellite
365
In our case the function
U, will be a periodic
funa-
tion with the period along the (100) axes being equal
L = (2M -t l)a = Q . ti. The structure factor for such a unit cell (the size of the complex is equal 2M&. see Fig. 1) can be expressed as : 2 F(J) exp (-&r&r’)
P,(S) =
~‘comp1ex = ;
F ;
I1p t
hF(%
Y>z)l
x exp [--~2n+&,
+
u,,,)l
x exp 1.--2~~~&
+
Uk.Jl
x
+ Un-,*)]
exp [-27ru,(Z,
where s,, slit s, are the components
FIG. 1. Model of tri-axial periodic modulated structure [only half of one “unit crll” of the periodic structure
let us neglect AF = 0).
divided by plane (010) is given].
data. (see Fig. I). This model may
also arise from consideration inkfaces.
But as will be shown
tangular modulation stages
of
in Fe-Be
decomposition
reflections
of
precipitation model
(or
be observed.
the interplanar
below,
of phase such
rec-
alloys appears at early
long
equilibrium may
spacing
before even
the
X-ray
intermediate)
According
to this
aIong each of these
changes so that at some part (LV -
2~1 of interplanar following
of late &ages of spinodal
as the result of sharpening
decomposition(a)
lattice,
pressed in units of ii*, F(J) is the structure the k unit, cell.
directions simultaneously
of the s vector
along t,he axes x, y3 z of t,he reciprocal
Oi.
/ IO101
spaces)
it is equal
part (2m of interplanar
For the simplicity the modulation it follows
cliagram for the Fe-Be composit,ion
of scattering
with experimental
at, 400°C t,wo phases arise. which
solution-nearly
“CuNiFe’
and “CuKiCo”
The
easy
= &4 per cent. For 450°C AFlp
we have still lower value: are nearly
differ
&-phase,
nearly 23 at. %
15 at’.% Be.
calculat’ion gives us AF/P values
states
al10ys,(~) as a result of the de-
ordered after the FeAl type, contains
These
of
power
from the metastable
from one another in content. of Be slightly: Be, a-solid
ex-
factor
of the calculation
This is in accordance
Indeed,
(1)
=
53
per
cent,.
equal to the values for the
alloys.
See also.(2)
For the whole crystal t,he scattering
a~ilplitude is:
t,o nl, at t,he
spaces) it. is equal to
repeated along the (100) (L2. This is periodically directions with the period of (2M + 1)s = L, where d is an average (100)interplanar lattice period).
The complex
spacing
(the average
shown in Fig. 1 is one
The equation for intensity
is :
repeating unit of such a periodic structure. The law of the change of atom coordiliates periodic structure
is given in the following
in tohis
equation: sin2 ns,QNz
X
Fe is a vector k-point
which characterizes
the position
of
of the mean lat,tice, U, is the atom displsce-
ment from k-point of the mean lattice; are the components
of this displacement
x, y, Z, Fk and U, are measured
u&,~.up,*, uk.% along the axes
in units of 8.
In
accordance with our supposition of the modulation of the interplanar spacing along three (100) axes simultaneously, components of ukuk,%, uk,?, and uk,, depend only on 2, y and z correspondingly, i.e.
sin2 nsZQN,
_._... .
sin2 T$~Q
sin2 TS,&
(2)
where N,. N, and N, are the numbers of periods of the modulakd
structure
along the axes x. y. z.
equation (2) follows that bhe value I(S) maxima
in the points of reciprocal
From
= /A( s)i2 has
space, for which
csx= h -1: (n/Q),where n is a whole number characterizing the order of satellites, and h is an order of reflection of the main maximum. It follows from equation
(2) the tri-axial
modulated
cause
lattice network
structure
space
the
must
appearance
of
in the the
periodic reciprocal
complicated
of satellite reflect.ions with coordinates
not,
TIAPKIS
AND
JIJ31’TI:
THI-AXIAL
t’ERI0DI.C
only {+*8JQ1 0: 0) as is usually observed, j&m,/&,
I&/&>
O> and
For the determination
modulat,ed
!I%/&).
of the intensity of these satellite the value of F,
with the help of equation
(1).
For this purpose the
of the sums of the following
type
structure
IS
must
be done:
observed
of interplanar
the whole complex
by 2M, cz is a mean lattice param-
eter, a, and a2 are spaces between
the planes
{loo]
outside and inside of the central part of t,he complex. As is seen from t*he model constant part
(Fig. l), these spaces are
along the direction
of the
summation
complex
(lOO> inside the central
(nz) and outside
(n,).
Upon
we get:
reflections
are
At this later stage we
rather diffuse reflections
which correspond
onal (with period
aT1 = a1 and cT1 = oa and aT2 =
a2 and cT2 = n,) conjugated of these
regions.
observed
A model
pictures
posed (the one tohat fully coincides Following, (*) let, us denote the number
stage with the
satellite
to two cubic (with periods a, and a,) and two tetrag-
interpretation
spaces in the central part of the complex by Xm, and in
367
XLLOYS
the so-called
where
seen has been studied in detail.
reflections it is necessary to calculate calculation
STRUCTURE
later stage t,hat follows
but
(I!I?~,,/Q, jl~z,/Q.
MOI)‘I’LATED
for
was pro-
with the model in
Fig. 1). As is seen from equations of the satellite reflections know the following
(3), (4) for the calculation intensity
parameters:
it is necessary
to
Q = 2M, 2m, 6, a,
and a2. Value Q = 2M (the period of the modulated struct’ure) is determined t,o the distances
from X-ray
data (according
of t,he satellite reflections
from the
main refle~t,ions) and from elect,ron-microscopic
data.
We note at once that these data are in satisfactory agreement.
The size of the complex
central part 2m
for Fe---Be alloys is simply estimated from the broadening of superlattice metastable
reflections
from the nuclei of the
phase with the C&l structure
which pre-
cipitate
in these alloys at the early stage.
position
of these superlattice
be determined.
reflections
The value ii. is determined
positSion of the main reflections the whole alloy) lattice. For the value of the scattering which
reflections A(s)
correspond
amplitude
to the maxima
we shall have the equation:
parameters
of our model
parameters
determined
of two compositions;
which
(4)
by equations such as
.y, = h + (n/Q,. We now use t)he above given equations of t-he satellit,e reflections
the &age of t,wo cubic and two tetragonal
up to the parameters t’he later stage-the
intensit,y for those correspond
to the alloys
alloy No. 1 with 19.5 at ‘A Be
are adequate
stages with intermediate we suppose
In this work the
scatt,ering
(the
increasing
of
30 2 30 2
1
1 1
2 -.-
min hr min hr
are given.
2.809 2.809 2.817 2.817
A
parameters
changes from the beginning
period at different~ stages of in t#he
phases which This change
(x1 and CL%.Thus
t,hat the whole process of the structural
these alloys was studied in Ref. 2.
aging and t,he manner of satellite disposition
to
reflec-
occurs so quickly that we do not usually observe the
ray diagrams
space of the matrix
aI and uZ tha,t correspond stage where independent
to our model are observed.
The structure of
lattice
phases.c2J
The same phenomeI]on takes place for value a2, i.e. the
and alloy No. 2 with 16.5 at. ok Be.
reciprocal
The latter
value is close to the value derived for the later titage-
tions from two cubic and two tetragonal for the cal-
while stTtdying Fe-Be
values of the modulation
(for
From the values & a2, 2N and
t,he d~eolnposit,ioxl at the initial stages quickly proceeds
= IE’I N1N,1L’,ct;,(s,)~,(~~)~~(s,)
where sZ, s, and s, are determined
culation
at the
of sat’ellite
from the
of the average
2n/, a value of a, can be easily calculated. points
From the
value aa ean
of the appearance
on ,X-
of the rather intense effects of diffuse satellite the
reflections)
modulat,ion
period
(and the value 2nc correspondingly) changes of parameters aI and a,.
2.76
140
2.76
200
2.76 2.76
80 1%
consists
in the
Q = 2%’ + 1
without significant The justice
of this
48 G 44 -____
x 12
ACTA
36X
METALLURGICA,
TABLE 2. The parameters
VOL.
of the calculated
19,
19’71
models of t,he modulated
structure 2m
Model
B C Z) ti: F
-
2.837 2.837 2.837 2.837 2.816 2.823
2.809 2.809 2.817 2.817 2.809 2.809
A
supposition intensities
is proved
2.7623 2.760 2.7670 2.7637 2.7966 2.7842
0.02659 0.02741 0.02485 0.02602 0.00691 0.01381
also by the calculation
of the satellite
reflections.
after
No.
(After
quality (alloy No. 2). In these cases the estimation of a the 10 step scale (10 steps are the intensity
1 and No.
values
2 (compare
of 2M,
treatments
obtained
at 400” for 30 min and
for 2 hr.
calculated
the effects
are revealed
in the form
most
clearly.)
2m, ii and a2 determined
experimentally
for these
(see Ref. 2) are given in
Table 1. The parameters of the calculated of different
the intensity
models are
satellite reflections
of the corresponding
nates are given with respect
main maximum) coordi-
to the position
of the
main maximum.
The results intensity
of the
(referred to
are given in Table 3. The satellite reflections corresponding
of The
given in Table 2. The results of the calculation intensity
Ss is seen from Table 4 satisfactory
1 and 2)
reflections
of the relative
of satellites (with respect to the correspond-
ing main maximum)
between
the
results.
measurements qualitative
are given in Table 4.
For alloy
of intensity
taken into account).
In other
more or less reliable quantitative microphotometer-either
were made with a
cases we could
not do
measurements
because
with a
of the nearness of
satellites to the main reflection and their incapability to be revealed
on
microphotograms
annealing at 400” for 2 hr, reflection of the absence of microcrystals TABLE
3. The relative intensity Reflection
[alloy
No.
and experimental table.
(IlO)] or because
of a sufficiently
high
of such measure-
agreement between theoretical
for the treatment
of alloy No. 2
of the intensity
X-ray
diagrams
However
that
the
it is easy to see in
intensity
of
satellites
(l/Q, l/Q, 0) is a little less than the intensity lites (l/Q, i/Q, 0) [or (i/Q, l/&,0)];
Let us here estimate contribution
The same occurs for alloy the quant,it,y of the possible of the scattering power
in the diffuse scattering intensity, above.
which we neglected
We shall calculate the intensity of the satellite [*
l/Q, 0, 0] of the reflection
model A (Table 2), providing cent (see above). equation
of satel-
this agreeswiththe
at 400” for 30 min.
of the modulation
that AP/P
In accordance
(1) F(z, Y, 2) = F,(z)
Reflection
200 for the = 14
+ F,(Y) + F&L (3), (4)
200, satellite coordinates
($&O)
(&OPO)
(&&O)
(&&O)
A B C ;
20 58.9 4.6 19.7 2.5
13 32.2 1.7 8.5 1.08
4.2 34.7 0.21 0.063 3.8
1.7 10.3 0.03 Kll
2.7 b:k
P
10.5
6.95
1.1
0.48
0.72
0.027 1.6
($o,o) 15 1890
(;,o,o) 165 4020
per
with our model in
for different models by equations
110, satellite coordinates
(;&O)
of
satellite (l/Q, l/Q, 0) and (l/Q, i/Q, 0) the value of less
(0,&O)
Model
only agree-
data is better than this is given in the
For example,
of satellites Za/Zhkl in per cent calculated
($o,o) (o,;,o)
not
quantitative
at 400” for 2 hr for the values
reflexions 1,
possible
the
exact
For the cases of visual estimation
the received qualitative
results of the calculation.
200 these measurements
were
and
where
ments and a large number of possible errors must be
No. 2 after the treatment
microphotometer.
data
cases
ment was achieved (the complication
110 and 200 for the treatment reflection
those
but also satisfactory
No. 1 after annealing at 400” for 30 min for reflections at 400” for 2 hr for
agreement was
experimental
For
than 1 step is given.
of the measurements
of the
tnain reflection).
Tables
these treatments
satellite
37.6 36.1 28.6 27.3 36.1 31.1
relative intensity was made visually and expressed by
the models for which the parameters
annealing
18 26 8 12 26 26
of the
are nearly the same as the parameters of structures of alloys
48 72 28 44 72 72
These results
will be given later. We calculated
138 205 82 127 205 205
($,o,o) 75 1020
(;,o,o) 44 1520
P,,
TABLE
_.-_
4. The comparison of the relative intensity of the satellite reflections I,/1 hkl in per cent or in 10 step scale* measured for alloys No. 1 and No. 2 and calculated in accordance with models A, B, C and D -._Reflection 200, satellite coordinates Reflection 110, satellite coordinates .._P -. -
(:3’0’0)(o,$o) (o,~~O)($o:oj ($;?o) (&go)
Tr~$y Alloy No. 1 400” 30 min Model A Alloy h’o. 1 400” 2 hr Model B Alloy x0. 2 4ooy 30 min Model C Alloy No. 2 4o0° 2 hr Model D
25
18
8
6
3
20 5b
13 4b
4.2 Pb
1.7 2b
2.7 26
58.9 tlb
32.2 SCIb
34.7 0
4.6
1.7
0.22
46 19.7
2b 8.5
lb 3.8
* At visual estimation
10.3 0 0.03
<
= (P -b- 3AF,fq,‘yy’yz’
+ (p + 3AF,)
x pl,$”. c&” . q&” --I-(E + 2AJ’, + AF,) x (q&l * P)vf* 47211 + f?Ji * 9;
* fPzf
+ qz’ . p; . rp,‘) + (P + AF, + zAF,t x (91,’ * 9y” ’ ye” + q&” * f&/’ * vz” + 9%” * hU . 9%‘) In our case (AF/P
1Q 0 0.08
i lb 1.6
($o,o) ($,o,o) (&o,o)
IQ
so
63
31
15 2000
165 3300
15 1300
44 1000
1890 -
4020 -
1020 -
1520 .-
-
-
I-_
-
-
-
_-
-
10 steps (lob) me the intensity of the main maximum.
F,, Fa depending only on x, y and z, respectively. In the center of the complex Fl(x) = F, = con&, outside of the center of the complex F,(x) = F, = con& The same is true for F, and F,. Upon summation in equation (l), taking into account (3), we get: F,(s)
(&!$‘O) ($‘&O) &o,o)
0%
= 1-4 per cent) AF,/P
= -AF,,J
E = 0.0133. Using the parameters of the model A (Table 2) we get for relative intensit,y of satellites (i/Q, 0, 0) and (l/Q, 0, 0) (in per cent of main reflexion) the quantities 169 and 74.5, respectively. In the approximation AF = 0 we get the quantities 165 and 75, that differ from the latter quantities not more than by 2 per cent. This is higher than the error of measurements of X-ray reflection intensity. Thus our approximation AF = 0 is correct enough. Thus, t*his study proves the correctness of the proposed model of the modulated structure for the case of the Fe-Be alloys. But formerly this model”) was earlier proposed for such alloys as Yiconal” and “CuNiFe,” for which the observed picture of diffuse
140 130 120 ,\" II0 1 IOQ-
h go? eo-Y 70 6050-
202224
9x
262830
10-3
3234
L,
Fra. 2. The change of the relative intensity of the various types satellite reflections of reflection (a) L, and (b) Au/&
8
Si
(110) depending on
scattering differs from that, for the Fe-Be alloys (for alloys of “ticonal” and “CuNiFe” types t,he socalled “cross” satellites, i.e. satellites with coordinates (l/Q, l/Q, O), (l/Q, l/Q, l/Q) and others, are not observed). The calculat,ions reported in the present paper allow us to explain this seeming disagreement. Let, us study within our model the change of the relat8iveint,ensity of satellites depending on the magnitude of the modulatio1~ period L = Q - 6 and the degree of modulation Aa@ = (a, - a&Z. In Fig. 2(a) the curves of dependence of the relative intensity of different satellite reflections of reflection (110) on the magnitude of the modulation period with given ratio 2Ml2m -_- 2.77 and An/a = 0.0267 are shown. It is clear that w&h small magnitude of modulat,ion period L only satellites in directions (100) from the main maximum, i.e. (l/Q, 0, 0) type. have noticeable inter&y. By increasing the modulation period, the intensity of bicross” satellit’es increases. When the magnit#udeof the modulation period is large enough (1; > 200 a, Fig. 2) some satellites [e.g. (l/Q, l/Q, O)] can become more intensive than satellites (l/Q, 0: 0), i.e. than those ones which are in direction {lOO>, and even more intensive than t,he Bragg maximum. This regularity is confirmed (as is seen from Table 4) by the studied Fe-Be alloys. These curves have the same character at other paramet’ers 2M/2n7 and Aa/& Thus, at small magnitudes of t!he modulation period, the intensit.y of “cross” satellites is so small that it is practically impossible t,o reveal them. The second important factor which changes the intensity ratio among various satellites is the degree of the modulation parameter Aa/& The dependence of the relative intensity of various types of satellites on this parameter is given in Fig. 2(b), where 2M/2m and L are constant parameters. They are equal t’o 2.77, 200 8, respectively. As is seen from Fig. 2(b), by increasing AajZ the intensity of the ‘icrosst’ sat,ellites increases faster than the “main” satellites (the satellites in the directions {loo)), so that with Aa/d = 0.0320 their intensity is comparable to the satellite intensity in directions (100) and with Au/E = 0.0330 the intensity of some “cross” satellites exceeds the intensity of the “main” ones. These curves explain why “cross” satellites may not be observed in such alloys as “ticonal”, “CuNiFe” and in other alloys with a modulated periodic structure. It is known that for these alloys the value of Aa/% does not exceed 0.010-0.015 and ratio 2M/2m usually varies from 2 to 4, i.e. near to value 2.7, used for the calculation of the curves given in Fig. 2(b). As is easily seen from this figure the intensity of the LLcross”
satellites relative to t~he intensity of the “main” satellites (satellites in the directions !lOOj); with values Aa/& in the given range of AajE values and even with such a modulation period as L = 200 A. will not’ exceed 2 or 3 per cent. It is necessary to conclude about’ t,he connection of the initial stage of decomposition of Fe-Be alloys (the so-called &age of modulated structure) with the stage of itlt,ermediate phases. As is seen from our investigations these stages differ chiefly only in thrb magnitndc of the modulaGon period. Indeed, with a sufficiently large value of &. t,he int’ensity of satellites increases so [see for example, Fig. 2(a)] that, it becomes much larger than t,he intensity of the main Bragg maximum. Wit,h further increase of the modulation period, i.e. with t,he &crease of the regions that compose the modulated stru(~t,~lre (two cubic and two tetragonal. see Fig. 1) the picture of satellites changes gradually into the picture of independent reflections from these regions. For the stage of modulated structure such independent’ reflect,ions can be seen also on the reflections w&h valnt~ XX2 that. are large enough.(2) CONCLUSION
The calculation of t,he diffuse scattering of X-rays with the help of the proposed model of tri-axial modulated periodic structure with periodic modulation of interplanar spacing and t,he scattering power along three directions {l@O) simult.aneously was carried out. The calculation showed that within t,he reciprocal lattice of such a cryst.al near the main (Bragg) maxima a complex spatial net of additiona, satellite reflections must arise, i.e. the satellite reflections removed from the main maxima not only in directions (100) but also in directions (llO), (111’: and others must appear. The quantitative measurement*s of t.he intensity of the satellite reflections for Fe-Be alloys with 16.5 and 19.5 at.% Be after annealing at 400” confirm the results of the calculation, proving t,he correctness of the chosen model. The calculation of the intensity of the various satellite reflections as a function of such parameters of the model as the modulation period L and the modulation degree Aa~~, allowed us to establish t,hat by increasing L and Aa~~ the intensity of t,he so called “cross” satellites (i.e. satellites in the directions (1 lo), (111) and others) at first is small in comparison with the intensity of the “main” satellites (in the directions (loo)), then it increases very fast so that under the definite values of L and Aa/d it may even exceed the intensity of the “‘main” satellites. These
TI.~I’KIS
ASD
sJI.Hl"L'I: THI-ASIAI,
PERIODIC
M01)L‘LA’l’El~
with modulated periodic structure (such as “CuNiFe”. “ticonal’” and others) the intensity of t,he “CuNiCo”. b‘cross” satellit>es must be negligible in comparison with t#he “main” satellites. Thus, during the experi-
ments rev~alcd
only
sat,ellit,es in directions
and tha,t is what, is ohserwd.
1100) should
bc
IS
ALLOYS
371
REFERENCES
cal~~llat,ions show &at, wit,h values L and &z/6 t,hat. correspond to the cases of the alloys studied previously
STRCCTlTRE
1. Kf. 3. %iIIK&Iil.11. 1‘. Em?
~~)Oltl~H~OBa --;SJtIicc:cP
178,
{lOcii).
2. AM.N.
,&KHC,YTA, K). ;J. Tmtmi- -tipmcTannorpa@wi, 13, 307 (1968). 3. V. DANIEL and H. LIPSON, Pm. R. Sot.18lA, 368 (1943). 4. HI. +!I,. THIIKHH. M. II. I’eopr~ee. M. R. ~Y$R@vTI~ GAlI CCCP 188, 80 (19cis).
5. J. W. CAHB, 24cta Met. 10, 179 (1962). 6. M. E. HARGREAVES. Aeta crystallogr. 4, 301 (1951). 7. M. \'.HEIMEXDAKL 11963i.
tend U. HET.BNER, Acta &let.11, 1119