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Local atomic arrangements in partially ordered CoPt3—I. studies of diffuse X-Ray scattering
LOCAL
ATOMIC ARRANGEMENTS IN PARTIALLY ORDERED STUDIES OF DIFFUSE X-RAY SCATTERING* H.
BERG,
JR.7
and
J. B.
CoPt,-I.
COHEN:
Diffuse scattering was measured in partially ordered CoPt, with a long-range order parameter (S) of 0.82. This scattering was separated into components due to local order and effects due to etomlc displacements, along three lines in reciprocal space. The equations for this linear separation are presented. The local order parameters were then determmed by solution of simultaneous equations. Objective tests of computer simulations of local atomic arrangements employmg these parameters were developed. It 1s shown that the experimental data indicates that misplaced atoms are arranged predominantly m small antiphase regions on {loo} planes, similar to partmlly ordered C$Au. ARRANGEMESTS
ATOMIQUES LOCAUX DANS ETUDE PAR DIFFUSION
CoPt, PARTIELLEMENT DES RAYONS X
ORDO?\‘SE-I.
La diffusion des rayons X a QtB mesurb dans un elhage CoPt, pertiellement ordon& (Paramktre d’ordre & longue distance S 0,82). Les auteurs ont dbtermm6 les composantes, dues it l’ordre local et aus effets de d6placements atomiques. sulvant, trois rang&s de l’espace r&xproque: 11s presentent len d’ordre local ant, Bt6 obtenus ensuite par Qquations permlttant cette dlvislon lin&ure. Les peram&es rCsolution d’&quations simultan&s. Des essais objectifs de simulation des arrangements atomiques locaux Les r&ultets expbrimentaux indiquent que B partir de ces parambtres ont BtB effectubs it l’ordinateur. les atomes non ordon&s sont rbpartis principalement dens de petits domaines antiphases sur les plans {loo}, comme dens Cu,Au partiellement ordon&. DIE
LOKALE ATOMANORDNUNG IN TEILWEISE GEORDNETEM UNTERSUCHUNG DER DIFFUSION RONTGENSTREUUNG
COP&---I.
Die diffuse Rantgenstreuung wurde in teilweise geordnetem CoPt, (mit einem Fernordnungsparameter S = 0,82) gemessen. Die Streuung konnte in zwei Komponenten zerlegt werden: Streuung an lokal geordneten Bereichen und Streuung eufgrund von Atomverschiebungen in drei Richtunpen des rezlproken Gitters. Die Gleichungen fti diese lineare Separation werden mitgetedt. Durch Losung simultaner Gleichungen wurden dann die lokalen Ordnungsparameter bestimmt . Es wurden objektive Tests der Computersimuletlon der lokalen Atoma~ordnung anhand dieser Parameter entwickelt. Die experimentellen Deten zeigen, daO die aus ihren Lagen verschobenen Atome iihnlich wie m teilweise geordnetem Cu,Au vorwiegend in klemen Antiphasenbereiche auf {loo)-Ebenen liegen.
INTRODUCTION
For
concentrated
applicable of local
solid
in most’ systems atomic
studies
conditional
one pair
is the
of X-rays can
obtain
probability,
the
procedure
for quantitative
configurations
of diffuse scattering such
solutions,
or neutrons. information
P$f,
that
is a B atom
interatomic
at the end of this
vector
From on
is a/2 (I + ne + n) for
the
of the probability
arranged. a[,,,,,
systems
short-range
with
is
generally
number
the
a body
pairs
can
Similarly
number
for
quantity
is
wit,h C
and N the total of AB
the number
be calculated.
this
the
and
NCX,P~f,
of at,oms. is the number
given shell.
systems,
positive
order,
The quantit’y
as the shell co-ordination
vector
if the atoms
For clustering
first-neighbor
usually negative.
if there
vector;
is the atomic fraction of B atoms,
which is the value are randomly
examination
is an A atom at t’he origin of an interatomic there
studies
In this equation X,
of dA,
Unfortunat’ely,
pairs for a
BB, BA such
in-
or face centered cubic crystal with one atom per lattice point. The values of 1, nL, n are integers. This
formation is not very specific concerning the appearance of the local atomic arrangements in the
probability
alloy, i.e. the actual shapes, sizes and compositions
is obtained
diffuse scattering, displacements particular parameter
by Fourier
inversion
aft’er contributions
(static and dynamic)
are separated.
the Warren short-range order is obtained from this inversion : a Irnn = 1 -
of the
due to atomic In
(a,,,)
B atoms,
P-W 2.
ACTA
METALLURGICA,
VOL.
21, DECEMBER
(1)
1973
such as a tetrahedron
of nearest-neighbors
the probability of a given configuration can be estimated,“) or the range of this probability can be calculated.(21
B * Received February 5. 1973; revised April 19, 1973. i Formerly Department of Meterlals Science, The Technologlcal Institute. Northwestern Umversity, Evanston, Illinois, now with Motorola, Inc., Phoenix, Arizona. : Department of Materials Science, The Technological Institute, Northwestern University, Evanston, Illinois.
of
specific local regions. Recently, techniques have been developed which allow such information to be obtained.(‘*2) For a particular small group of A and
For larger groups computer
simulation
has been employed, .(2) A and B atoms selected at random from a large three-dimensional array are interchanged, until a specified number of the Warren parameters are satisfied. Then the configuration is examined. While many tests have been made to 1579
1580
ACT.4
METALLURGICA,
VOL.
21. 1973
assure that the final configurations do not change where h,, h,, h3 am continuous co-ordinates in redrastically uit,h different starting states, or with ciprocal space and : different a[,,,,, from a given alloy (e.g. the first three, (II)) F lmn = %Jin - &7w or first, fourt,h and fifth, etc.) there is still a question as to how closely the computer simulations match 4mn = ~~X,~7&32 for E, 733,n = 2q the actual atomic arrangements in the alloy. (where q is an integer), (1 c) There is another tool suitable for examining the 0 (Id) local atomic arrangements in alloys-namely, the almn = --1*X 0 ABX S2 for 1 + n? -+- n = 29. Fieid-Ion Microscope (FIM). Unfortunately, it is The degree of long-range order (Sj is defined by: useful w&h only certain alloys and, for concentrat,ed alloys, only when long-range order is present. In this study a comparison of these two approaches is made using the alloy CoPt,. In Co-Pt alloys examined in the FIM, the Co atoms are effectively invisible,(3*4) The unit cell employed here is that of the Ll, strucand good images are obtained when order is not quite ture-a cubic cell with Co at corners and Pt at faces. complete. For a partially ordered Co-Pt alloy, if The term r, is the fraction of GC(face) sit,es correctly occupied. Note that plooofor a stoichiomet’ric R,B the atoms on wrong sublattices are not randomly alloy is 1 - S2; this term represents the total diffuse arranged, a modulation of the diffuse X-ray scatscattering due to local atomic arrangements in a tering results, and this can be readily detected because unit cell in reciprocal space in an alloy, in Laue of the large difference in atomic number of the monotomic units per atom. It is jnde~ndent of species. the modulations produced by the actual arrangements ; There are other reasons for studying this system. that is, the other q,,,,, merely modulate this term The ordered structure has the Strukturbericht and are all zero if the misplaced atoms are randomly designation Ll,,(5’ like CusAu. In the fully ordered distributed. [This diffuse intensity is in addition to state there are Co atoms at the corners of a cubic the superstructure peaks which are due to the average unit cell and Pt atoms at face centers. Similar state of order and have intensity proportional to s?.] structures are important hardening agents in Ni base In this study quantitative measurements were made alloys. Many detailed studies of ordering kinetics, only along [hOO], [hhO] and [hhh] lines in reciprocal equilibrium order, ant,iphase t,opography, etc. have space. Along each of these lines IsRo may be written been made, but primarily with Cu,Au. It is generally assumed that other such struct,ures will behave in a as a simple series. For example, along an hhk line, if Vz = 0 for i > 8 (where i = (E2 + m2 + ,z2)/2): similar way. But a recent study”J) has shown that the order is much lower at T, in CoPt, than in CusAu, I SRO=PzoA,ll co9 2?rph,, (3R) that there is a two-phase region close to stoichiometry where : and that during annealing, (1OOj antiphase domain boundaries do not, predominate, as they do for CusAu. A:” = ~0 + f3p, + 6rp, + $43 + .t+,, (3b) Thus, further research on the behavior of alloys A:” = ‘3931 + $7, + 4~3 + 12~5 + 697, + 1Q,. with this struct,ure is needed. In this first part,, a description is given of studies (3c) concerning diffuse X-ray scattering, and the atomic A;” = 6973+ 69~ + 12~5 + 129, + 69s. (3df configurations obtained from comput’er simulation All’ = 2p, + 129%. 3 Gel are presented. In the second paper, studies with the FIM will be discussed. Inversion of Isno obtained along a minimum repeat length yields the As’s. There are six addnional ‘“A” coefficients along [hhO], and five along [&Of. The SCATTERING THEORY relationships between these and the 9’s in t.hese other two directions can be found in Ref. 8. Previously, In the presence of long range order, the scattering only the [AOO]and [iLhO]directions were employed, due to the local arrangements of misplaced atoms requiring the assumption that yesa = 1 - s2 t,o (Is,,) may be written:“) solve equations like 3b-3e for the va’s from the A,‘s. I SRO = NX,X,(f, -fd" However, this total integrated intensity of local order (all) is often in considerable error, including cos 29r(Eh,+ mh, + nh,), (la) x ~IzZ%~ uncorrected background, multiple scattering, terms m n
BERG
AND
COHEX:
ATOMIC
ARRAAKGEMENTS
due to atomic displacements not adequately corrected for. etc. By adding the third line presented here, 15 equations relating the A,‘s to nine p?*‘s result, and a least-squares solution for all Q)*‘s is possible. This was first demonstrated for b.c.c. systems.‘g) It, is apparent from equations 3 that the cp’s must, fall t,o near-zero values for ypl, i 2 8 for this met,hod to be valid. If higher order v’s were significant ,. satellites might be present along directions other than the measured ones. To examine this possibilit,y, before the linear measurements were performed. an examination was made of t,he diffuse intensity in a volume in reciprocal space; no additional detail was det’ected. Prior to invert’ing the data obtained along these linear samples of reciprocal space, the effect’s of abomic displacements must be removed. The procedures for doing this are described in the Appendix. EXPERIMENTAL
METHODS
The alloy was prepared by Engelhard Industries as a i in y; 1 in x 0.1 in rolled strip. Chemical analysis from one end indicated a composition of 74.89 at’. *A Pt:, 25.00 at.% Co and the other end, 75.16 at. ‘A Pt, 24.73 at. % Co. Unce~ainties in this analysis thus appear to be w 0.1 at. %. (The numbers do not add to 100 per cent.) Spectrographic analysis for 1’7 elements gave a total impurity content of 0.01 wt. %. principally Rh. A quenched powder specimen gave a lattice parameter which when converted to Kx units and compared to the work of Gehhardt’ and KosteG”) indicated a composition of 74.3-74.5 at. %. The composition was therefore judged to be within a few tenths of a per cent of st,oichiometry. From part of this strip single crystals were grown by Metals Research, Ltd. employing the Bridgeman method, in an alumina crucible under gettered Argon. Several attempts were required and the alloy was inverted each time. One small crystal e 0.5 in. in diameter was obtained, as well as a coarse grained 0.5 in. diameter rod, two inches in length. From lattice parameter measurements the composition was unchanged from the original material, but, there was a gradient of w 1 at. ‘A over t,he length of the two-inch rod. A slice was cut from the small crystal to a plane halfway between the (100) and (110) planes. A slice from the rod was taken near the stoichiometric end, near a (100) plane. These were polished by standard metallographic procedures, with a final t,reatment consisting of alternately polishing and etching in hot aqua regia until no further sharpening of Laue spots was observed. The
IX
PARTIALLY
ORDERED
CoPt $---I
f381
total mosaic in the slice from the small crystal was 0.6“; it was this crystal that was employed in quantitaGve measurements of diffuse scattering and the other will be referred to as the companion crystal. Surface roughness was measured by counting the fluorescent, Co&(ll) produced by MoK, radiation. A Xi filter reduced the PtL, intensity during this measurement, while a further reduction \vasachieved with a pulse-height, anal_yzer and balanced FeO-MnO fibers in t,he diffract.ed beam. Surface roughness decreased t’he COB, fluorescence by * l:! per cent at 30” 20, 10 per cent at 40” % and * 6 per cent at 80” 28. All data were corrected for this factor. Checks were made of the effect of crystal orient’ation on this correction; none were found. AS the data was taken w&h Co&;,, the correction was made dire&l> t,o the measurements of diffuse scattering. The crystals were encapsulated in quartz at a vacuum of IO-? Torr or bet,ter and given the following heat treatment : 77O”C-3 h, 69O”C-i
days, G83”C-12
days,
67O”C-7 days, 665”C-30
dars -5 days, .* fI.&Q”C---~~ 62O”C--9 days, 600°C--4 days. 58O’Cd days, 560°C---4 days, 55O”C-4
days,
54O”C-2 days, 52O”C-3
days, 5(KPC--4
48O”C-8
days, 45O”C-10
days, 46O”C-i
days, days,
7 days at 44O”C, 430°C. 42O”C, 410°C. 400°C for 16 days and i days at’ 385’C. The crystals were t’hen re-etched slightly, to remove a m 25A layer of pure Pt (due to evaporation). (This was det,ected in preliminary studies of diffuse s~tt~ering.) The heat t~at,rnen~ was designed to achieve a degree of long-range order Q 1 and a large anti-phase domain size, following the kinetic studies in Ref. 6. From the integrated intensities of Bragg peaks of the companion crystal, S was det,ermined to be 1.010(25). (See Ref. G for details of this determination.) Measurements of t,hediffuse scattering were then made along the necessary lines on the primary fully ordered crystal. This crystal was then encapsulated again, and heated to 640°C for four hours and quenched into ice water. The degree of long-range order (S) was determined to be 0.82. The separated short-range order intensity (ISR,,) after the first’ heat treatment (fully ordered condition, thus this separated data was only the spurious effects) were subtracted from those for S = 0.82, to correct for any spurious effects such as fluorescence, multiple Bragg scattering, tails of Bragg peaks, etc. (The separated intensity within 0.2 reciprocal lattice
ACTA
1582
METALLURGICA,
units of superstructure peaks for the fully ordered crystal was reduced by S2 before the subtraction.)* All measurements of diffuse scattering were made with a modified G. E. XRD-5 diffract,ometerc’2) with Picker elecbronics. The incident CoK, beam (5OkV, 7ma) was first monochromated with a pyrolitic graphite crystal bent to focus in the vertical plane at the receiving slits. Soller slits that focused at the receiving slits were employed in the exit port of the monochromator. Total horizontal and vertical divergences were 1.7’ and 2.8’ respectively. Counting was carried out for a fixed number of counts in a monitor count,er, which measured the VK, fluorescence from a thin Vanadium oxide foil in the exit beam from the monochromator. In this way fluctuations due to voltage, current and barometric pressure were minimized. The specimen was held in a special goniostat with a hemispherical Be cover, 0.030 in. thick, mounted on a G. E. quarter circle.(13) The device was evacuated to 5 p or less. The system was automated with a Digital Equipment Corporation PDP 8-L computer with software designed for this work.‘14) The data was corrected for dead time. Air scatter and electronic noise (M 4 per cent of the data) was measured with a beam trap in place of t,he specimen. The amount of PtL, fluorescence was determined by measuring the intensity of the diffracted beam with and without an absorbing Ni foil (1, I,, respectively) at several points in reciprocal space. The transmission (T) of this foil for PtL, and CoK, had previously been determined : I[(1 -
X) T,,
+ X Tp,] = I,i,
(4)
where X is the fraction of the total intensity due to PtL, fluorescence. This fluorescence was typically 15 per cent of a measured intensity. The direct beam’s intensity was determined from an Al powder compacted at 25,000 psi with the data on scattering * During the course of the exp1orator.v studlea in a volume m reciprocal space mentloned above several problems were discovered, the firat of whtch were the powder rmga of the Pt film mentioned earlier. Secondly. separated SRO mtensiby near fundamental peaks, especially along hO0 du-ectlonn, was especially high. This was less pronounced for a fully ordered alloy than for the partially ordered allvy. Thw could be due to mosaic effects, mcreased by quenchmp a partially ordered crystal. Finally, there was a roughly constant background in ISRO: on Inverting the SRO data m a volume pOoO. whxh should be about 0.3, was 0.6. The difference. 0.3. 1s about the usual error reported currently in studws of local order; there are many plausile reasons for this: fluorescence from the specimen and its impurities. multlple acattermp, etc. But because of all these problems a difference methwl appears for supermr. Taking differences m a volume requires excessively long times in obtaining the data. Thus the analysis was carried out along lines. Actually. except for qOoO. p)pll and rp,,,. the q’s determined from the volume data were quite similar to those obtained from the trdmear analysis.
VOL.
21,
1973
factors, etc. from Refs. 15 and 16. Values from the 200, 220, 311 were averaged to give a value of 0.204 * lo* cps, with a spread of &O.OOi * 10s cph. RESULTS
AND
DISCUSSION
All d&a was placed on an absolute scale prior to separation and solution for the y’s. The sccttering fact,ors employed were t’hose of Cromer(17)j corrected for dispersion.“6) Compton scattering was calculated from the results in Ref. 18. It is difficult, to determine the polarization of a monochromator with the long7 wave length radiation employed in this study. Processing the data twice assuming first, an ideally perfect and t’hen an imperfect’ monochrornator changed the ~7~‘s by w 5 per cent,. A value for polarization halfway between these extremes was chosen for final processing. The results for the vi’s are given in Table 1. Foul different extrapolations under fundamental reflections caused only minor changes in these values; the averages are given. The data extrapolated to 100 and 110 superstructure refle&ions along the TABLE 1. Short-range
order parameters CoPt, S = 0.82 0.3155 -0.0402 0.0683 - 0.0066 0.0193 -0.0113 0.0044 0.0022 0.0116
in CoPt,.
S = 0.82
Cu& s = 0.8O”~ -0.037 + 0.034 $0.001 A 0.007 - 0.008 O.OOl, - 0.00” t0.012
different lines gave the same intensity, as should be the case. It is to be noted that v,,,,,, (= 1 - 49) should be 0.31 with possible error of f 8 per cent due to the error in S (see Ref. 6) and the error in the direct beam intensity; the actual result,s are within two per cent of this value. For comparison, the vi values determined previously for Cu,Au, S = 0.80 are also given in Table 1. The pattern and the magnitude of the values is quite similar, except for 4)200which is twice as large for CoPt,, The results for Cu,Au were employed previously in computer simulations to obtain the atomic configurations. It was found that the misplaced atoms were primarily in plate-like anti-phase domains on (lOO} planes, containing w 4-5 atoms per domain. There were also some domains on (111 j planes, containing w 20per cent of the misplaced atoms. From these simulations and others we have carried out’20) the larger qZoo for CoPt, indicates that the domains may be larger in this system than in Cu3Au. To
BERG
examine
these details.
simulations
with the data obtained The
simulation
reviewed.@)
ARRANGEMENTS
were carried
in some initial
have
recently
been
of “atoms”
are
state in a computer
periodic
IX
An
PARTIALLY
initially
changing
large numbers
a three dimensional
out
in this study.
procedures
Briefly,
constructed
ATOMIC
COHEN:
AND
array.
in
A and B at,ome
within
random
array
CoPt
was
altered
0.5 per cent of the desired
Various
is denot’ed
by
the
values.
a simulation
and
compared.
to the atomic arrangements the almn corresponding in t#he simulated crystal agree with a set of input (measured) values. At, this point the configurations
computer
runs, these were not, utilized
t,o more
closely
are plotted the
past,
simply
on low index this
analysis
lookmg
for
planes and examined. was usually
predominant
performed
features,
small regions of a specific configuration them.
More quantitative
procedures
in this work to minimize such examinations, of this
study,
one can be influenced
simulate
only the (110) planes were analyzed of
as
analyzed
this
study, only
the on
with
simulations
ideas as to what possible
mistaken as to the habit of some region.
to
be
For example,
domains lie on (110) planes
planes
in
[[llo]]
or
[[lOO]]
directions and appear as rods on these planes. If the regions are small. a random misplaced atom or two nearby of
a small
a small rod will give the appearance
two-dimensional
such
in that portion
“maps”
planes
and
were only
also
for
Pt,
One Search run was also made with S = 0.80 and
is made
It is sometimes
(100)
computer
data for Part II
all
qlrnn = 0.00,
six
arrangement inserted in
platelet
on
the
(100)
that
is
for
a
near-random
of misplaced atoms. All regions were a synthesis on mixed species planes
(planes with 50 per cent Co, 50 per cent Pt at,oms counts
intersect’
Also,
The tests employed are illustrated in Fig. 1, along with the vi’s from the Lattice and Search runs.
easily
they
studies.
at’oms.
in the fully ordered
if small planar antiphase
t,he FIM
of
in counting,
as only the Pt atoms are visible in the FIJI and as
should be present, or by the regions or clusters most recognized.
periodic
of
nature
of computer
by previous
(hkl).
in both types
In
were developed
a comparison
result’s from the FIM. In examining the outputs
are employed
by
and counting
the subjective
and to provide
where
such
Such a
While
conditions
and interchanged
intcr-
counts were then made on both a synthesis
boundary
at random
by
t’erm Search
until
are then chosen
1583
$-I
atoms until the first six alphas and S \vere
simulation and
ORDERED
state).
are given in Tables
above, this counting (110)
planes,
Rods
could
required
Results
to contain
one
branched
were examined.) atom
at least three atoms
along [[lOO]] or [[llo]] directions counted. Two dimensional regions terms of misplaced
(As indicat,ed
was done only on mixed-species
but all six variants have
for t,he various
2 and 3.
but
were
in a row
in order to be were defined in
Pt atoms connect’ed by a,, [[lOO]]
plane being examined. A series of tests was initiated. a specified particular inserted
number
size. shape, at random
computer.
For a given test,
of small antiphase
regions
of a
and habit
were
at,om model
in a
composition
in a 4,000
The value of long-range
was
adjusted
such
regions;
at, random
using
t’hose atoms
a sufficient on
wrong
order (S = 0.80) not
number sublat,tices.
s_ynthesis of this kind will be specified Lattice (hk2), implying
this study {loo}.)
developed
A computer by the term
program for this purpose
in Ref.
to allow printouts
The
in
distributed
that regions were inserted on
(The computer
(hk2) planes. was initially
included
was
13 and modified
in
on planes other than
first six short -range
order
coefficients
(not the P)~‘s. but x,‘s) corresponding to t’he Lattice (hlcl) s.ynthesis were determined as part, of the program. It is t,o be emphasized
t,hat, this program
is ?lot a
simulat,ion to match measured parameters. It is simply a way of obtaining local order parameters from a model of an alloy. These six values were then employed in another program, a simulation in which t’he atoms were arranged to satisfy the alphas.
FIG. 1. The various regions were inserted in a 4,000 atom model of CoPt,. The regions shown were inserted at random in the number given. The composition of the surrounding material was adjusted to yield CoPts and b = 0.80; 0 = Pt, 1 = Co. The first six v’s resulting from thlri synthesis are referred to as Lattice syntheses. The first SIX a’s (not the q’s) were then employed in a (Search) simulation in which 4,000 atoms were forced to match S, the a’s and the composition. The first six p’s after this simulation are given in the last SW rows (q, and a are related by equation 1 in the text).
ACTA
1584 T.4BLE
2.
(a)
Distribution
Test
of misplaced
21,
1973
in rods a;d?Ot;odimensional
regions
in the computer
Fraction of misplaced Rods along
tests; A,B
in regions 2-D regions 3 atoms >7
alloy,
atoms
Number of combined atomst + 900’
201
0.223
0.090
0.050
0.037
0.000
490 419 454 424 236 200 483 695
0.545 0.465 0.504 0.470 0.262 0 .N__ *o-7*7 0.536 0.770
0.256 0.169 0.145 0.125 0 .0”” __ 0.034 0.055 0.147
0.07s 0.062 0 077 0:o;i 0.214 0.113 0.011 0.017
0.163 0.171 0.234 0.19; 0.020 0.034 0.439 0.560
0.049 0.052 0.088 0.050 0.010 0.009 0.103 0.273
577 496 379 429
0.641 0.552 0.42 1 0.469
0.358 0 .__.,.,*, 0.112 0.109
0.094 0.030 0 097 0:053
0.16s 0.209 0.234 0.231
0.070 0.087 0.131 0.034
Large domains Lattice(100) Search(100) Lattice( 110) Search(ll0) Distribution
A atoms
VOL.
Total number of combined? atoms
Random Small domains Lattice(100) Searoh( 100) Lattice(ll0) Search(ll0) Lattice(ll1) Search(ll1) Lattice(3-D) Search(3:D)
(b)
METALLURGICA,
of only combmed
misplaced
A atoms
[1(~00?11
in rods and in 2-D regions Rods
Test
>
UC1lo)11
in the computer
along
[[(I lOi11
[I(lWll
tests:
.-l,B
atoms
alloy,
8 = o.811
2-D regions 3-3 atoms :-7 atoms
Random Small domams Lattice(100) Search( 100) Lattioe( 110) Search(ll0) Lattice(ll1) Search(ll1) Lattice(3.D) Search(3-D)
0.510
0.283
0.207
0.000
0.504 0.420 0.348 0.339 0.086 0.189 0.110 0.203
0.160 0.155 0.089 0.127 0.823 0.622 0.022 0.024
0.336 0.425 0.564 0.547 0.091 0.189 0.868 0.775
0.101 0.130 0.212 0.151 0.046 0.049 0.206 0.384
Large regions Lattice( 100) Search(100) Lattice’( 116) Search(ll0)
0.578 0.484 0.304 0.275
0.152 0.065 0.061 0.136
0.271 0.452 0.635 0.588
0.113 0.18H 0.357 0.096
* A total of 900 misplaced Pt atoms were present in any one computer simulation. t Only in this column does combmed atoms include 2-D regions with three misplaced
TABLE
3. Fraction
of combined
Pt atoms.
misplaced A atoms in domains onmised {llO) planes in the computer requirements of the (110) masks. d,B alloy, S = 0.80
4 atom (mm. of 4)
6 atom (min. of 5)
9 atom (min. of 6)
tests meeting
12 atom (min of 7)
the minimum
16 atom (min. of 8)
Random Small domains Lattice(100) Searoh( 100) Lattice( 110) Seamh(ll0) #I Search( 110) #2 Lattioe(ll1) Searoh(ll1) Lattice(3-D) Search(S-D)
0.06
0.06
0.12
0.07
0.04
0.033 0.162 0.176 0.115 0.180 0.017 0.020 0.590 0.410
0.051 0.160 0.200 0.066 0.170 0.021 0.025 0.266 0.396
0.086 0.163 0.240 0.103 0.230 0.076 0.060 0.337 0.347
0.119 0.160 0.200 0.081 0.238 0.003 0.350 0.240 0.313
0.118 0.210 0.100 0.138 0.137 0.072 0.040 0.131 0.243
Large domains Lattice( 100) Search(lO0) Lattice(ll0) Search(ll0)
0.062 0.113 0.222 0.196
0.097 0.125 0.296 0.177
0.118 0.127 0.436 0.173
0.106 0.189 0.308 0.191
0.104 0.231 0.331 0.103
BERG
COHES:
AND
and a, [[llo]]
vectors,
ATOMIC
ARRANGENEKTS
and were required
IN
t’o contain
PARTIALLY
fraction
are in antiphase
more than three atoms and more than one branching
when the
at’om to
compared;
they
The items
counted
distinguish
“combined”
such
atoms
are
and t(wo-dimensional in Table
3 were
regions
those
The
intended
to
sensitive
to both shape and size of antiphase in Table
two dimensional of such
a
possible, e.g. (i.e. two
along
masks
4-atom
mask
along
[[lOO]].
was required t’o contain of antiphase atom
Pt
plane
as
regions,
as square
included
[[llo]]
indicated
masks
three
directions
were
in the scanned
as
i
(2) (111)
No
result’s
in Tables
2 and
3 clearly
between
the models introduced
syntheses
and the
simulations
adds
additional
literature as these
support
that’ tests
previous ones.
such
to
are
st’ringent
than
This is because ordered
TABLE 4. (8) Distribution
Run ::
already
simulations
are more
at,oms are in the
that
runs).
less than
cases
of misplaced
meaningful,
those
misplaced
of the
in
larger
of
are present
with
rat’io of the
cent, are
more
than
fractions
of
than those in [[llo]]. result in many
more
atoms
randomly
arranged,
are present
in [[lOO]]
regions
of misplaced
of 3 atoms
for all obher tesbs,
atoms
or more
except
platelem
because
or combined
[[llo]]
rods
or
combined
than
the
is
atoms w 3, atoms
in the other
can be distinguished rat,io of the
tesm.
fraction
in [[lOO]] rods to
and in
t#he fraction [[lOO]]
of
rods
Notme, however,
syntheses:
Frection Rods
[t(1oo)ll
CoPt, S = 0.80, qlmnnfrom experiment of misplaced
t[( 1lo)11
0.185 0.167
Pt atoms m 2-D regions >3 atoms >7 atoms 0.155 0.227
0.035 0.047
Run
[[(loo)11
1 2
0.495 0.379
Run
6 etoms (mm. of 5)
1 2
0.049 0.089
0.069 0.092
of combmrrl
[[(llo)ll 0.093 O.lOti
Pt atoms meet,ing the requirements
4 atomb (mm. of 4)
(d) Fraction
Pt atoms
Rods
of total misplaced
misplaced
0.075 0.090
Run
6 atoms (min of 5)
1 2
0.132 0.200
0.185 0.209
2-D regions >3 atoms >7 atoms 0.413 0.515
0.202 0.204
of the (110) masks
Mask size 9 etoms (mm. of 6) 0.070 0.094
Pt atoms meeting the requuvments
4 atoms (mm. of 4)
is
that
* 810 IS the total number of misplaced Pt atoms in this run. (b) Distribution of combmed Pt atoms
(c) Fraction
is
for the
the fract,ions of atoms in either kind of rod is smaller
Pt atoms in computer
0.470 0.499
fract’ions
20 per
arrays
of atoms
and (110)
other
most
Number of combined atoms/810*
381 404
the
and only a small
of misplaced
Total so of combined atoms
from
most of the Co and Pt
matrix
atoms
rods, and the fraction
(4) (100)
This in
of
(111) platelets.
in Lattice
(Search
details
rods than in [[lOO]] rods.
in two-dimensional
indicate
have
Over
the
all misplaced
and [[llo]] much
The
Also.
only a small number
the
1) are
for all the test,s.)
combined
arrays.
anti-ph ase platelem
(3) with
output. agreement
each.
atoms in [[llo]]
(110)
across
of the
atoms in [[lOO]] rods is greater
number
is evident
the following
in two-dimensional
atoms
and
table.
(This
qIrn,, in Fig.
similar
revealed
1585
8-I
atoms in rods of 0.2 or less, while at least
quarters
present
a region
once on a given
from
are very
in two-dimensional
2 x 9 Co
To be counted,
more than
these
were
regions.
al,,,,, (calculated
combined
rods from
at least the minimum
atoms
was counted
are
the size or compactness
These atoms
sites
t,wo at,oms
2 distinguishes
regions-not
regions.
masks
CoPt
the atomic configurations : (1) small 3-D antiphase regions
compact
of
data
The
rods
described
distinguish
atoms.
The
all
masks
clusters while the
Pt
rods.
comprising
regions.
misplaced
from
ORDERED
16 atoms (min. of 8)
0.064 0.121
0.094 0.126
of the (110) masks
Mask s,ze 9 atoms (min. of 6) 0.188 0.212
12 8t,oms (mm. of 7)
12 atoms (mm. of 7) 0.172 0.273
16 atoms (min. of 8) 0.251 0.284
ACTA
1586
for (110) the
platelets
fraction
in
than two
for (100)
METALLURGICA,
platelets,
dimensional
but that
regions
greater
than three atoms is larger. (5)
the
masks
are
useful
in
distinguishing
3-D
VOL.
APPENDIX. SEPARATION OF 18~0 AND INTENSITY DUE TO ATOMIC DISPLACEMENTS
plane from those on another. In
Table
4 the
m.easured
some
results are
q’s
indication
of two
given.
of the
errors
size of the model, 4000 atoms. test results state
in Tables
of long-range
misplaced
simulations
The
involved
give
with
Comparison
the
with the
2 and 3 shows that in a partial order CoPt,
contains
ant,iphase with the matrix. Further
If up to square terms in displacements
of atoms from
lattice sit,es are included, t,hen the fatal diffuse intensity at a point, uritten’21~22)
h,, h,.&
in reciprocal
space,
can
be
most of the
atoms in the form of small (100)
as for C&Au.
with
differences
1973
21. C’. J. SPARKSand B. BORIE. Local dtornic Arranaeww,tt.~ Studies by X-ray Diffractiorb, p. 53. edited by J. B.“C~HE\ and J. E. HILLIARD. Gordon & Breach (1966); .4&r tryst. A27, 198 (1971). 22. J. E. GRAGC, JR. and J. B. COHES, dcta Net. 19, 507 (1971).
regions or random arrays from platelets. They are not very useful in distinguishing platelets on one
the
21,
platelets,
This is the same situation
discussion
will be postponed
ISRO
=
(hl,h,2, h3)
+
h&(hl.
h,.h,)
until the results in Part II are presented. ACKNOWLEDGEMENTS
This
research
was
Science Foundation.
sponsored
by
One of the authors
fully acknowledges
an NDEA
western University
Cabell fellowship.
the
+
~2QX(~,~~,~
h3)
+
h2%&l,
+
~&2~~,(~23~2.
+
h2.4)
u?_y(~,.h1.h,)
i-
hz2B_y(h2.h,.
h,)
National
(H. B.) grate-
fellowship and a NorthPortions
work were submitted
by H. Berg in partial
of the requirements western University.
for the Ph.D.
degree
of this
fulfillment at North-
The
Fourier
REFERENCES
f
h~,~,,(~,~
coefficients
4.
of the
detailed
they
vanish
equations
here (however,
see Refs.
W)
involving
pract,ically be it is necessary
at, the eighth
for them
h?).
terms
the displacements (Q, R, S) cannot obtained in linear analyses (because to assume
1. P. C. CLAPP,Critical Phenomena in Alloys, Magnets and Superconductors, p. 299. edited by R. E. MILLS. E. &HER and R. I. JAFFEE. McGraw-Hill (1971). 2. J. E. GRAOGI,JR., P. BARDHA~ and J. B. COHES, ibid, p. 309. 3. H. N. SOUTHWORTHand B. RALPH, Phil. ,Uag. 14, 383 (1966): 21,545 (1970). Appl. Phys. Lett. 9, 7 4. T. T. TSON~ and E. W. MiiLmR. (1966); J. appl. Phys. 28, 3531 (1967). _ 0. A. H. GEISLERand D. C. I~IARTI~,J. appl. Phys. 22, 375 (1952). 6. El. BERN and J. B. COHEN, Afet. Trawls. 3, 1681 (1972). (For actual Bragg intensities of the crystal employed in this study with S = 0.82 see H. BERG, Ph.D. thesis, Sorthwestern Umversity, 1972.) 7. D. R. CHIPMAN,J. appl. Phys. 27, 739 (1956). 8. L. H. SCHWARTZ and J. B. C’OHES,J. appl. Phys. 38, 598 (1965). 9. T. ERICSSOS,S. LI~DE and J. B. COHES, J. appl. Cryst. 4, 31 (1971). 10. E. GEBHARDTand W. KOSTER, 2. &fetaZZfi. 32,253 (1940). 11. P. M. DE WOLFF, Acla Cyst. 9, 682 (1956). 12. L. H. SCHWARTZ,L. A. MORRISOS and J. B. COHEN,ddv. X-ray Analyeia 7, 281 (1964). 13. J. E. GRAOQ, JR. Ph.D. thesis, Northwestern University, 1970. 14. T+LRICHESSON,L. MORRISONand Ii. PAAVOLA, J. appl. Cry&. 4, 524 (1971). 16. B. W. BATTERMAN,D. R. CHIPMAXand J. J. DEMARCO. Phys. Rev. 122, 68 (1961). M. J. COOPER,Acta Crysl. 10,1067 (1963). ;:: D. T. CROMERand D. T. MANN, LASL Report LA-3689 (1967). D. T. CROMER,J. Chem. Phye. 50,4857 (1964). ::: P. C. GERLEN and J. B. ~OHES, J. appl. Phys. 40, 5193 (1969). 20. P. C. GEHLEN, Ph.D. thesis, Northwestern University, 1966.
h)
shell),
so bhe
n-ill not, be presented
21 and 22).
It is sufficient
to point out that Q is a sum over In/)/of sine functions, R and S involve
whereas
cosine
functions.
Q is a
series in terms of mean static displacements atoms from the average lattice sites.
of
Qs involves the sum over 1~01 of the X components of t’he displacements atomic
vector
of atoms separat’ed
lmn.
Since
the
by an inter-
mat,erial
is
cubic,
Qs (hlh2hs) = Q, (h2hlh3) = Q, (&h&J. because the displacements (X,,,) = ( Ynrln) = ::Xn,,llJ. Thus, 0111) Qs is required in equation Al : Q, and Q, do not appear, and similarly square
(static
includes
cross
for the t,erms R, S. and
d_ynamic)
products
of
R involves
displacements displacements
meanand
such
S as
(XrmnYlmn). Each of these terms has the symmetry shown in Table Al, which results in a minimum repeat volume in reciprocal must be obtained space.
Because
variations
space over which the component
to know
it throughout,
of these symmetries,
of the terms in equation
minimum volume required smaller than a unit cell.
for
reciprocal
and the different (Al)
with h, the
measurement
The separation of t’he terms along the three of interest will now be examined. Consider that term
is lines first
Isno (h,, 0,2) = Isno (h,, 0, 0), because this has a periodicity of 2 in reciprocal space.
BERG
ASD
COHEN:
ATOMIC
ARRANGEMENTS
IN
PARTIALLY
ORDERED
CoPt,-I
1X7
Al. Symmetries of the components of the total measured diffuse intenslt,y, for an f.c.c. material. All components hare a period of two in h,.
TABLE
Symmetry
Term h, = 0, h, =
ISRO
&$
(i =
h, + h, = 1, h, -
across
1, 2, 3),
h, = 1, h, = -h,
h, = 0, h, = 0, h, = *, h, = 4, h, = i-h,, h, * k, = 1,
Q,
h1 = 0, h, = 1 (whole line is antisymmetric) h, = 0, h, = +&, (i = 1, 2, 3),
x, S”.
11, = kh,,h,
& hs = 1
h, = 0, A, =
+&, h, = iha,
h, 3~ h, = 1,
h, = 0, h, rf 4, h3 = 0, h, = 54 h, = 0, h, = h, (i, j = 1, 2, 3;
IT
(n-hole line is antisymmetric)
i = j)
AeAilT(hl, 0, 2) = I,@,.
Then : I&
0,54 = Isa&,.
(42) + w,(~,,o~
+ w&,
h,* 0) + h?R&,
-+ 4R,(2, h,. 0) + 2V,,(2,
0>2)
0,2) - I,(2
-
Employing
= line I - line 2 = 2Q!,(h,, 0,2) + 4% -
l)JUh,, 092) (A2b)
+ 4&!,(2: A,, 01, A&@,:
0,2) = I,(2
- h,. 0$2)
Jr I,(2
+ IL,.0.2) -
21&L,, 0,2)
= line 2 i_ line 3 - 2(line 1) = SR,(h,, 0,2).
(A2c)
The lines are shown in the minimum repeat volume in reciprocal space in Fig. Al.
the
I,(2
-
h,, 0,2)
symmetries
(A2d) in Table
Al,
S,,
J2J
= I,(O. 0, 2 :
-
(h,, h,, h3) = -S,, t---h,, h,, &); therefore S,, (2, I+$) must be zero for all h,. Similarly S,, (h,, 0, 2) = S,, (2, h,, 0) = 0, and Q, (2, h,, 0) = 0. However, because there are errors in the actual separation due to statistical errors in any experiment these terms have therefore been included. All terms in equation A2a except Q, (2, h,, 0) and R, (2, h,, 0) are obtained from oquation (A2b-A2d.) Keeping in mind the symmetry in Table Al and the periodicity of two, it can be seen that R, (2, h,, 0) = R, (0, 0,2 - h) and Q, (2, h,, 0) = Q, (0, 0, 2 - IL,). Therefore, the remaining terms can be obt,ained from the following differences : A,I&O, 0, 2 -
“1
$2)
= PAS.&, 0, 2).
(A2a) h,, 0,2)
I&,,
- k,, 2,2)
= line 1 + line S - line 7 - line 2
A,, 0)
i ~~(O)~_~~(~~, 0, 2)* A1.z&,, 0,2) = I&,.
-
2)
0,2) + I,(2
h,) -
I&,
0,2 -
h,)
= line 4 - line 5 = “Q,(O, 0, 2 - n,) - 4(O)R,(O, 0,2 :
+ 2f2 - h&Y,,{2 i
A&(0.0,
2-
-
= 2 (line 5) = 8R,(O, 0,2 -
’ “2
(A24
n,)
= 21,(2.0,2
Fm. Al. The mmimum repeat volume in reciprocal space to separate mt,ensity due to focal order from intensity due to atomic displacements m an f.c.c. alloy. The numbered linear segments permit the separation along the h,, 0, 2 lines.
h,, O,O),
h,)
h,) -
21,(0,0,
2-
h,)
2 (line 4) h,).
Wf)
All size terms can therefore be evaluat,ed from measurements along eight. lines, three of which overlap, in a compact region of reciprocal space. [It is this attempt at compactness and to avoid the origin of reciprocal space that has led to the choice of the volume and hence the lines involved.] With
ACTA
1588
~ETALLURGI~A,
the size terms and equation (A2a,) Isno (hO0) can be evaluated. A similar analysis can be employed to determine ISno (h, h, 0), which is equal to lsRo (h,, 0, 2 - h,). All terms but S,, (2 - h,, n,, 0) are determined from A,l&h,, 0, 2 - h,) and from A,*fr(h,, 0, 2 - h,). The term &‘,,(I&,,h,, 2) = -S&2 - h,, h,, 0) and the former can be obtained, for h, = 0.0 - 0.5, from
VOL. 21, 1973 synthesized on a computer. (The required equat,ions for the size terms are in Ref. 21, 22.) At,omic displacements in Q, R. S were calculated for first TABLE A2. Comparison of the short,-range order parameters obtained from the trilinear analysis wzth the oragmai Input values
From analysis of synthesized data
Input
= I&h,, h,, 2) + I,(2 - I@,. -
I,(2
2 -
-
h,, 2 - h,, 2)
h,, 2) - I&h,, 2 -
h,, 2)
h,, h,, 2)
= as,, h,, A,, 2), end for h, = 0.5 &41&l, = 1241 - I,(1
(Asa)
-
h,, 1) + I,(1
+ hr. 1 + h,, 1)
h,, 1 + h,, 1) - I,(1
= PS(,,,h,% 4,2).
(from Ref. 8)
0 1 2 3 4
0.360 -0.037 0.034 0.001 0.007
5
-0.008
O.fOOi -0.009
0.000 - 0.002 0.012
- 0.003 0.000 0.015
6 7 8
1.0:
&2) h,, 1 -
i
+ h,, 1 -
h,, 1) (A3b)
Finally, consider the h,, h,, 2 - h, line, equivalent to h, h, h and extending from 0, 0, 2 to 0.5, 0.5, 1.5. Forming AJ, rend A,21, allows the determination of all needed quantities, when the symmetries in Table Al are considered. One of the major reasons for the attempt to make the measuring region as compact as possible (and for including the S,, terms along [hOO]) is that the size terms include scattering factor ratios. OnIy if these ratios are con&ant in the measuring region are the periodicities and symmetries strictly maintained. To test this trilinear procedure, data was
0.360 - 0.04 1 0.03ti ct.002
neighbors from the sizes of the atoms and were allowed to decrease, following the inverse-square law, with increasing interatomic distance. The scattering variation was allowed. Compton modified scattering was included as well as a. variation in surface roughness. The local order coefficients employed were those measured from Cu,Au. S = 0.8O.(n’) A direct beam was chosen to give 2000 counts per point. (Wit’h the monochromat,ic Co& radiation employed in this experiment’ the actual counting time would be 20 min per point.) A normally distributed random error was then added t,o the intensity at each “measured” point. The initial values of ?i and these values aft‘er analysis of this simulated intensity are shown in Table A2. The method appears adequate.
ATOMIC ARRANGEMENTS IN PARTIALLY ORDERED STUDIES OF DIFFUSE X-RAY SCATTERING* H.
BERG,
JR.7
and
J. B.
CoPt,-I.
COHEN:
Diffuse scattering was measured in partially ordered CoPt, with a long-range order parameter (S) of 0.82. This scattering was separated into components due to local order and effects due to etomlc displacements, along three lines in reciprocal space. The equations for this linear separation are presented. The local order parameters were then determmed by solution of simultaneous equations. Objective tests of computer simulations of local atomic arrangements employmg these parameters were developed. It 1s shown that the experimental data indicates that misplaced atoms are arranged predominantly m small antiphase regions on {loo} planes, similar to partmlly ordered C$Au. ARRANGEMESTS
ATOMIQUES LOCAUX DANS ETUDE PAR DIFFUSION
CoPt, PARTIELLEMENT DES RAYONS X
ORDO?\‘SE-I.
La diffusion des rayons X a QtB mesurb dans un elhage CoPt, pertiellement ordon& (Paramktre d’ordre & longue distance S 0,82). Les auteurs ont dbtermm6 les composantes, dues it l’ordre local et aus effets de d6placements atomiques. sulvant, trois rang&s de l’espace r&xproque: 11s presentent len d’ordre local ant, Bt6 obtenus ensuite par Qquations permlttant cette dlvislon lin&ure. Les peram&es rCsolution d’&quations simultan&s. Des essais objectifs de simulation des arrangements atomiques locaux Les r&ultets expbrimentaux indiquent que B partir de ces parambtres ont BtB effectubs it l’ordinateur. les atomes non ordon&s sont rbpartis principalement dens de petits domaines antiphases sur les plans {loo}, comme dens Cu,Au partiellement ordon&. DIE
LOKALE ATOMANORDNUNG IN TEILWEISE GEORDNETEM UNTERSUCHUNG DER DIFFUSION RONTGENSTREUUNG
COP&---I.
Die diffuse Rantgenstreuung wurde in teilweise geordnetem CoPt, (mit einem Fernordnungsparameter S = 0,82) gemessen. Die Streuung konnte in zwei Komponenten zerlegt werden: Streuung an lokal geordneten Bereichen und Streuung eufgrund von Atomverschiebungen in drei Richtunpen des rezlproken Gitters. Die Gleichungen fti diese lineare Separation werden mitgetedt. Durch Losung simultaner Gleichungen wurden dann die lokalen Ordnungsparameter bestimmt . Es wurden objektive Tests der Computersimuletlon der lokalen Atoma~ordnung anhand dieser Parameter entwickelt. Die experimentellen Deten zeigen, daO die aus ihren Lagen verschobenen Atome iihnlich wie m teilweise geordnetem Cu,Au vorwiegend in klemen Antiphasenbereiche auf {loo)-Ebenen liegen.
INTRODUCTION
For
concentrated
applicable of local
solid
in most’ systems atomic
studies
conditional
one pair
is the
of X-rays can
obtain
probability,
the
procedure
for quantitative
configurations
of diffuse scattering such
solutions,
or neutrons. information
P$f,
that
is a B atom
interatomic
at the end of this
vector
From on
is a/2 (I + ne + n) for
the
of the probability
arranged. a[,,,,,
systems
short-range
with
is
generally
number
the
a body
pairs
can
Similarly
number
for
quantity
is
wit,h C
and N the total of AB
the number
be calculated.
this
the
and
NCX,P~f,
of at,oms. is the number
given shell.
systems,
positive
order,
The quantit’y
as the shell co-ordination
vector
if the atoms
For clustering
first-neighbor
usually negative.
if there
vector;
is the atomic fraction of B atoms,
which is the value are randomly
examination
is an A atom at t’he origin of an interatomic there
studies
In this equation X,
of dA,
Unfortunat’ely,
pairs for a
BB, BA such
in-
or face centered cubic crystal with one atom per lattice point. The values of 1, nL, n are integers. This
formation is not very specific concerning the appearance of the local atomic arrangements in the
probability
alloy, i.e. the actual shapes, sizes and compositions
is obtained
diffuse scattering, displacements particular parameter
by Fourier
inversion
aft’er contributions
(static and dynamic)
are separated.
the Warren short-range order is obtained from this inversion : a Irnn = 1 -
of the
due to atomic In
(a,,,)
B atoms,
P-W 2.
ACTA
METALLURGICA,
VOL.
21, DECEMBER
(1)
1973
such as a tetrahedron
of nearest-neighbors
the probability of a given configuration can be estimated,“) or the range of this probability can be calculated.(21
B * Received February 5. 1973; revised April 19, 1973. i Formerly Department of Meterlals Science, The Technologlcal Institute. Northwestern Umversity, Evanston, Illinois, now with Motorola, Inc., Phoenix, Arizona. : Department of Materials Science, The Technological Institute, Northwestern University, Evanston, Illinois.
of
specific local regions. Recently, techniques have been developed which allow such information to be obtained.(‘*2) For a particular small group of A and
For larger groups computer
simulation
has been employed, .(2) A and B atoms selected at random from a large three-dimensional array are interchanged, until a specified number of the Warren parameters are satisfied. Then the configuration is examined. While many tests have been made to 1579
1580
ACT.4
METALLURGICA,
VOL.
21. 1973
assure that the final configurations do not change where h,, h,, h3 am continuous co-ordinates in redrastically uit,h different starting states, or with ciprocal space and : different a[,,,,, from a given alloy (e.g. the first three, (II)) F lmn = %Jin - &7w or first, fourt,h and fifth, etc.) there is still a question as to how closely the computer simulations match 4mn = ~~X,~7&32 for E, 733,n = 2q the actual atomic arrangements in the alloy. (where q is an integer), (1 c) There is another tool suitable for examining the 0 (Id) local atomic arrangements in alloys-namely, the almn = --1*X 0 ABX S2 for 1 + n? -+- n = 29. Fieid-Ion Microscope (FIM). Unfortunately, it is The degree of long-range order (Sj is defined by: useful w&h only certain alloys and, for concentrat,ed alloys, only when long-range order is present. In this study a comparison of these two approaches is made using the alloy CoPt,. In Co-Pt alloys examined in the FIM, the Co atoms are effectively invisible,(3*4) The unit cell employed here is that of the Ll, strucand good images are obtained when order is not quite ture-a cubic cell with Co at corners and Pt at faces. complete. For a partially ordered Co-Pt alloy, if The term r, is the fraction of GC(face) sit,es correctly occupied. Note that plooofor a stoichiomet’ric R,B the atoms on wrong sublattices are not randomly alloy is 1 - S2; this term represents the total diffuse arranged, a modulation of the diffuse X-ray scatscattering due to local atomic arrangements in a tering results, and this can be readily detected because unit cell in reciprocal space in an alloy, in Laue of the large difference in atomic number of the monotomic units per atom. It is jnde~ndent of species. the modulations produced by the actual arrangements ; There are other reasons for studying this system. that is, the other q,,,,, merely modulate this term The ordered structure has the Strukturbericht and are all zero if the misplaced atoms are randomly designation Ll,,(5’ like CusAu. In the fully ordered distributed. [This diffuse intensity is in addition to state there are Co atoms at the corners of a cubic the superstructure peaks which are due to the average unit cell and Pt atoms at face centers. Similar state of order and have intensity proportional to s?.] structures are important hardening agents in Ni base In this study quantitative measurements were made alloys. Many detailed studies of ordering kinetics, only along [hOO], [hhO] and [hhh] lines in reciprocal equilibrium order, ant,iphase t,opography, etc. have space. Along each of these lines IsRo may be written been made, but primarily with Cu,Au. It is generally assumed that other such struct,ures will behave in a as a simple series. For example, along an hhk line, if Vz = 0 for i > 8 (where i = (E2 + m2 + ,z2)/2): similar way. But a recent study”J) has shown that the order is much lower at T, in CoPt, than in CusAu, I SRO=PzoA,ll co9 2?rph,, (3R) that there is a two-phase region close to stoichiometry where : and that during annealing, (1OOj antiphase domain boundaries do not, predominate, as they do for CusAu. A:” = ~0 + f3p, + 6rp, + $43 + .t+,, (3b) Thus, further research on the behavior of alloys A:” = ‘3931 + $7, + 4~3 + 12~5 + 697, + 1Q,. with this struct,ure is needed. In this first part,, a description is given of studies (3c) concerning diffuse X-ray scattering, and the atomic A;” = 6973+ 69~ + 12~5 + 129, + 69s. (3df configurations obtained from comput’er simulation All’ = 2p, + 129%. 3 Gel are presented. In the second paper, studies with the FIM will be discussed. Inversion of Isno obtained along a minimum repeat length yields the As’s. There are six addnional ‘“A” coefficients along [hhO], and five along [&Of. The SCATTERING THEORY relationships between these and the 9’s in t.hese other two directions can be found in Ref. 8. Previously, In the presence of long range order, the scattering only the [AOO]and [iLhO]directions were employed, due to the local arrangements of misplaced atoms requiring the assumption that yesa = 1 - s2 t,o (Is,,) may be written:“) solve equations like 3b-3e for the va’s from the A,‘s. I SRO = NX,X,(f, -fd" However, this total integrated intensity of local order (all) is often in considerable error, including cos 29r(Eh,+ mh, + nh,), (la) x ~IzZ%~ uncorrected background, multiple scattering, terms m n
BERG
AND
COHEX:
ATOMIC
ARRAAKGEMENTS
due to atomic displacements not adequately corrected for. etc. By adding the third line presented here, 15 equations relating the A,‘s to nine p?*‘s result, and a least-squares solution for all Q)*‘s is possible. This was first demonstrated for b.c.c. systems.‘g) It, is apparent from equations 3 that the cp’s must, fall t,o near-zero values for ypl, i 2 8 for this met,hod to be valid. If higher order v’s were significant ,. satellites might be present along directions other than the measured ones. To examine this possibilit,y, before the linear measurements were performed. an examination was made of t,he diffuse intensity in a volume in reciprocal space; no additional detail was det’ected. Prior to invert’ing the data obtained along these linear samples of reciprocal space, the effect’s of abomic displacements must be removed. The procedures for doing this are described in the Appendix. EXPERIMENTAL
METHODS
The alloy was prepared by Engelhard Industries as a i in y; 1 in x 0.1 in rolled strip. Chemical analysis from one end indicated a composition of 74.89 at’. *A Pt:, 25.00 at.% Co and the other end, 75.16 at. ‘A Pt, 24.73 at. % Co. Unce~ainties in this analysis thus appear to be w 0.1 at. %. (The numbers do not add to 100 per cent.) Spectrographic analysis for 1’7 elements gave a total impurity content of 0.01 wt. %. principally Rh. A quenched powder specimen gave a lattice parameter which when converted to Kx units and compared to the work of Gehhardt’ and KosteG”) indicated a composition of 74.3-74.5 at. %. The composition was therefore judged to be within a few tenths of a per cent of st,oichiometry. From part of this strip single crystals were grown by Metals Research, Ltd. employing the Bridgeman method, in an alumina crucible under gettered Argon. Several attempts were required and the alloy was inverted each time. One small crystal e 0.5 in. in diameter was obtained, as well as a coarse grained 0.5 in. diameter rod, two inches in length. From lattice parameter measurements the composition was unchanged from the original material, but, there was a gradient of w 1 at. ‘A over t,he length of the two-inch rod. A slice was cut from the small crystal to a plane halfway between the (100) and (110) planes. A slice from the rod was taken near the stoichiometric end, near a (100) plane. These were polished by standard metallographic procedures, with a final t,reatment consisting of alternately polishing and etching in hot aqua regia until no further sharpening of Laue spots was observed. The
IX
PARTIALLY
ORDERED
CoPt $---I
f381
total mosaic in the slice from the small crystal was 0.6“; it was this crystal that was employed in quantitaGve measurements of diffuse scattering and the other will be referred to as the companion crystal. Surface roughness was measured by counting the fluorescent, Co&(ll) produced by MoK, radiation. A Xi filter reduced the PtL, intensity during this measurement, while a further reduction \vasachieved with a pulse-height, anal_yzer and balanced FeO-MnO fibers in t,he diffract.ed beam. Surface roughness decreased t’he COB, fluorescence by * l:! per cent at 30” 20, 10 per cent at 40” % and * 6 per cent at 80” 28. All data were corrected for this factor. Checks were made of the effect of crystal orient’ation on this correction; none were found. AS the data was taken w&h Co&;,, the correction was made dire&l> t,o the measurements of diffuse scattering. The crystals were encapsulated in quartz at a vacuum of IO-? Torr or bet,ter and given the following heat treatment : 77O”C-3 h, 69O”C-i
days, G83”C-12
days,
67O”C-7 days, 665”C-30
dars -5 days, .* fI.&Q”C---~~ 62O”C--9 days, 600°C--4 days. 58O’Cd days, 560°C---4 days, 55O”C-4
days,
54O”C-2 days, 52O”C-3
days, 5(KPC--4
48O”C-8
days, 45O”C-10
days, 46O”C-i
days, days,
7 days at 44O”C, 430°C. 42O”C, 410°C. 400°C for 16 days and i days at’ 385’C. The crystals were t’hen re-etched slightly, to remove a m 25A layer of pure Pt (due to evaporation). (This was det,ected in preliminary studies of diffuse s~tt~ering.) The heat t~at,rnen~ was designed to achieve a degree of long-range order Q 1 and a large anti-phase domain size, following the kinetic studies in Ref. 6. From the integrated intensities of Bragg peaks of the companion crystal, S was det,ermined to be 1.010(25). (See Ref. G for details of this determination.) Measurements of t,hediffuse scattering were then made along the necessary lines on the primary fully ordered crystal. This crystal was then encapsulated again, and heated to 640°C for four hours and quenched into ice water. The degree of long-range order (S) was determined to be 0.82. The separated short-range order intensity (ISR,,) after the first’ heat treatment (fully ordered condition, thus this separated data was only the spurious effects) were subtracted from those for S = 0.82, to correct for any spurious effects such as fluorescence, multiple Bragg scattering, tails of Bragg peaks, etc. (The separated intensity within 0.2 reciprocal lattice
ACTA
1582
METALLURGICA,
units of superstructure peaks for the fully ordered crystal was reduced by S2 before the subtraction.)* All measurements of diffuse scattering were made with a modified G. E. XRD-5 diffract,ometerc’2) with Picker elecbronics. The incident CoK, beam (5OkV, 7ma) was first monochromated with a pyrolitic graphite crystal bent to focus in the vertical plane at the receiving slits. Soller slits that focused at the receiving slits were employed in the exit port of the monochromator. Total horizontal and vertical divergences were 1.7’ and 2.8’ respectively. Counting was carried out for a fixed number of counts in a monitor count,er, which measured the VK, fluorescence from a thin Vanadium oxide foil in the exit beam from the monochromator. In this way fluctuations due to voltage, current and barometric pressure were minimized. The specimen was held in a special goniostat with a hemispherical Be cover, 0.030 in. thick, mounted on a G. E. quarter circle.(13) The device was evacuated to 5 p or less. The system was automated with a Digital Equipment Corporation PDP 8-L computer with software designed for this work.‘14) The data was corrected for dead time. Air scatter and electronic noise (M 4 per cent of the data) was measured with a beam trap in place of t,he specimen. The amount of PtL, fluorescence was determined by measuring the intensity of the diffracted beam with and without an absorbing Ni foil (1, I,, respectively) at several points in reciprocal space. The transmission (T) of this foil for PtL, and CoK, had previously been determined : I[(1 -
X) T,,
+ X Tp,] = I,i,
(4)
where X is the fraction of the total intensity due to PtL, fluorescence. This fluorescence was typically 15 per cent of a measured intensity. The direct beam’s intensity was determined from an Al powder compacted at 25,000 psi with the data on scattering * During the course of the exp1orator.v studlea in a volume m reciprocal space mentloned above several problems were discovered, the firat of whtch were the powder rmga of the Pt film mentioned earlier. Secondly. separated SRO mtensiby near fundamental peaks, especially along hO0 du-ectlonn, was especially high. This was less pronounced for a fully ordered alloy than for the partially ordered allvy. Thw could be due to mosaic effects, mcreased by quenchmp a partially ordered crystal. Finally, there was a roughly constant background in ISRO: on Inverting the SRO data m a volume pOoO. whxh should be about 0.3, was 0.6. The difference. 0.3. 1s about the usual error reported currently in studws of local order; there are many plausile reasons for this: fluorescence from the specimen and its impurities. multlple acattermp, etc. But because of all these problems a difference methwl appears for supermr. Taking differences m a volume requires excessively long times in obtaining the data. Thus the analysis was carried out along lines. Actually. except for qOoO. p)pll and rp,,,. the q’s determined from the volume data were quite similar to those obtained from the trdmear analysis.
VOL.
21,
1973
factors, etc. from Refs. 15 and 16. Values from the 200, 220, 311 were averaged to give a value of 0.204 * lo* cps, with a spread of &O.OOi * 10s cph. RESULTS
AND
DISCUSSION
All d&a was placed on an absolute scale prior to separation and solution for the y’s. The sccttering fact,ors employed were t’hose of Cromer(17)j corrected for dispersion.“6) Compton scattering was calculated from the results in Ref. 18. It is difficult, to determine the polarization of a monochromator with the long7 wave length radiation employed in this study. Processing the data twice assuming first, an ideally perfect and t’hen an imperfect’ monochrornator changed the ~7~‘s by w 5 per cent,. A value for polarization halfway between these extremes was chosen for final processing. The results for the vi’s are given in Table 1. Foul different extrapolations under fundamental reflections caused only minor changes in these values; the averages are given. The data extrapolated to 100 and 110 superstructure refle&ions along the TABLE 1. Short-range
order parameters CoPt, S = 0.82 0.3155 -0.0402 0.0683 - 0.0066 0.0193 -0.0113 0.0044 0.0022 0.0116
in CoPt,.
S = 0.82
Cu& s = 0.8O”~ -0.037 + 0.034 $0.001 A 0.007 - 0.008 O.OOl, - 0.00” t0.012
different lines gave the same intensity, as should be the case. It is to be noted that v,,,,,, (= 1 - 49) should be 0.31 with possible error of f 8 per cent due to the error in S (see Ref. 6) and the error in the direct beam intensity; the actual result,s are within two per cent of this value. For comparison, the vi values determined previously for Cu,Au, S = 0.80 are also given in Table 1. The pattern and the magnitude of the values is quite similar, except for 4)200which is twice as large for CoPt,, The results for Cu,Au were employed previously in computer simulations to obtain the atomic configurations. It was found that the misplaced atoms were primarily in plate-like anti-phase domains on (lOO} planes, containing w 4-5 atoms per domain. There were also some domains on (111 j planes, containing w 20per cent of the misplaced atoms. From these simulations and others we have carried out’20) the larger qZoo for CoPt, indicates that the domains may be larger in this system than in Cu3Au. To
BERG
examine
these details.
simulations
with the data obtained The
simulation
reviewed.@)
ARRANGEMENTS
were carried
in some initial
have
recently
been
of “atoms”
are
state in a computer
periodic
IX
An
PARTIALLY
initially
changing
large numbers
a three dimensional
out
in this study.
procedures
Briefly,
constructed
ATOMIC
COHEN:
AND
array.
in
A and B at,ome
within
random
array
CoPt
was
altered
0.5 per cent of the desired
Various
is denot’ed
by
the
values.
a simulation
and
compared.
to the atomic arrangements the almn corresponding in t#he simulated crystal agree with a set of input (measured) values. At, this point the configurations
computer
runs, these were not, utilized
t,o more
closely
are plotted the
past,
simply
on low index this
analysis
lookmg
for
planes and examined. was usually
predominant
performed
features,
small regions of a specific configuration them.
More quantitative
procedures
in this work to minimize such examinations, of this
study,
one can be influenced
simulate
only the (110) planes were analyzed of
as
analyzed
this
study, only
the on
with
simulations
ideas as to what possible
mistaken as to the habit of some region.
to
be
For example,
domains lie on (110) planes
planes
in
[[llo]]
or
[[lOO]]
directions and appear as rods on these planes. If the regions are small. a random misplaced atom or two nearby of
a small
a small rod will give the appearance
two-dimensional
such
in that portion
“maps”
planes
and
were only
also
for
Pt,
One Search run was also made with S = 0.80 and
is made
It is sometimes
(100)
computer
data for Part II
all
qlrnn = 0.00,
six
arrangement inserted in
platelet
on
the
(100)
that
is
for
a
near-random
of misplaced atoms. All regions were a synthesis on mixed species planes
(planes with 50 per cent Co, 50 per cent Pt at,oms counts
intersect’
Also,
The tests employed are illustrated in Fig. 1, along with the vi’s from the Lattice and Search runs.
easily
they
studies.
at’oms.
in the fully ordered
if small planar antiphase
t,he FIM
of
in counting,
as only the Pt atoms are visible in the FIJI and as
should be present, or by the regions or clusters most recognized.
periodic
of
nature
of computer
by previous
(hkl).
in both types
In
were developed
a comparison
result’s from the FIM. In examining the outputs
are employed
by
and counting
the subjective
and to provide
where
such
Such a
While
conditions
and interchanged
intcr-
counts were then made on both a synthesis
boundary
at random
by
t’erm Search
until
are then chosen
1583
$-I
atoms until the first six alphas and S \vere
simulation and
ORDERED
state).
are given in Tables
above, this counting (110)
planes,
Rods
could
required
Results
to contain
one
branched
were examined.) atom
at least three atoms
along [[lOO]] or [[llo]] directions counted. Two dimensional regions terms of misplaced
(As indicat,ed
was done only on mixed-species
but all six variants have
for t,he various
2 and 3.
but
were
in a row
in order to be were defined in
Pt atoms connect’ed by a,, [[lOO]]
plane being examined. A series of tests was initiated. a specified particular inserted
number
size. shape, at random
computer.
For a given test,
of small antiphase
regions
of a
and habit
were
at,om model
in a
composition
in a 4,000
The value of long-range
was
adjusted
such
regions;
at, random
using
t’hose atoms
a sufficient on
wrong
order (S = 0.80) not
number sublat,tices.
s_ynthesis of this kind will be specified Lattice (hk2), implying
this study {loo}.)
developed
A computer by the term
program for this purpose
in Ref.
to allow printouts
The
in
distributed
that regions were inserted on
(The computer
(hk2) planes. was initially
included
was
13 and modified
in
on planes other than
first six short -range
order
coefficients
(not the P)~‘s. but x,‘s) corresponding to t’he Lattice (hlcl) s.ynthesis were determined as part, of the program. It is t,o be emphasized
t,hat, this program
is ?lot a
simulat,ion to match measured parameters. It is simply a way of obtaining local order parameters from a model of an alloy. These six values were then employed in another program, a simulation in which t’he atoms were arranged to satisfy the alphas.
FIG. 1. The various regions were inserted in a 4,000 atom model of CoPt,. The regions shown were inserted at random in the number given. The composition of the surrounding material was adjusted to yield CoPts and b = 0.80; 0 = Pt, 1 = Co. The first six v’s resulting from thlri synthesis are referred to as Lattice syntheses. The first SIX a’s (not the q’s) were then employed in a (Search) simulation in which 4,000 atoms were forced to match S, the a’s and the composition. The first six p’s after this simulation are given in the last SW rows (q, and a are related by equation 1 in the text).
ACTA
1584 T.4BLE
2.
(a)
Distribution
Test
of misplaced
21,
1973
in rods a;d?Ot;odimensional
regions
in the computer
Fraction of misplaced Rods along
tests; A,B
in regions 2-D regions 3 atoms >7
alloy,
atoms
Number of combined atomst + 900’
201
0.223
0.090
0.050
0.037
0.000
490 419 454 424 236 200 483 695
0.545 0.465 0.504 0.470 0.262 0 .N__ *o-7*7 0.536 0.770
0.256 0.169 0.145 0.125 0 .0”” __ 0.034 0.055 0.147
0.07s 0.062 0 077 0:o;i 0.214 0.113 0.011 0.017
0.163 0.171 0.234 0.19; 0.020 0.034 0.439 0.560
0.049 0.052 0.088 0.050 0.010 0.009 0.103 0.273
577 496 379 429
0.641 0.552 0.42 1 0.469
0.358 0 .__.,.,*, 0.112 0.109
0.094 0.030 0 097 0:053
0.16s 0.209 0.234 0.231
0.070 0.087 0.131 0.034
Large domains Lattice(100) Search(100) Lattice( 110) Search(ll0) Distribution
A atoms
VOL.
Total number of combined? atoms
Random Small domains Lattice(100) Searoh( 100) Lattice(ll0) Search(ll0) Lattice(ll1) Search(ll1) Lattice(3-D) Search(3:D)
(b)
METALLURGICA,
of only combmed
misplaced
A atoms
[1(~00?11
in rods and in 2-D regions Rods
Test
>
UC1lo)11
in the computer
along
[[(I lOi11
[I(lWll
tests:
.-l,B
atoms
alloy,
8 = o.811
2-D regions 3-3 atoms :-7 atoms
Random Small domams Lattice(100) Search( 100) Lattioe( 110) Search(ll0) Lattice(ll1) Search(ll1) Lattice(3.D) Search(3-D)
0.510
0.283
0.207
0.000
0.504 0.420 0.348 0.339 0.086 0.189 0.110 0.203
0.160 0.155 0.089 0.127 0.823 0.622 0.022 0.024
0.336 0.425 0.564 0.547 0.091 0.189 0.868 0.775
0.101 0.130 0.212 0.151 0.046 0.049 0.206 0.384
Large regions Lattice( 100) Search(100) Lattice’( 116) Search(ll0)
0.578 0.484 0.304 0.275
0.152 0.065 0.061 0.136
0.271 0.452 0.635 0.588
0.113 0.18H 0.357 0.096
* A total of 900 misplaced Pt atoms were present in any one computer simulation. t Only in this column does combmed atoms include 2-D regions with three misplaced
TABLE
3. Fraction
of combined
Pt atoms.
misplaced A atoms in domains onmised {llO) planes in the computer requirements of the (110) masks. d,B alloy, S = 0.80
4 atom (mm. of 4)
6 atom (min. of 5)
9 atom (min. of 6)
tests meeting
12 atom (min of 7)
the minimum
16 atom (min. of 8)
Random Small domains Lattice(100) Searoh( 100) Lattice( 110) Seamh(ll0) #I Search( 110) #2 Lattioe(ll1) Searoh(ll1) Lattice(3-D) Search(S-D)
0.06
0.06
0.12
0.07
0.04
0.033 0.162 0.176 0.115 0.180 0.017 0.020 0.590 0.410
0.051 0.160 0.200 0.066 0.170 0.021 0.025 0.266 0.396
0.086 0.163 0.240 0.103 0.230 0.076 0.060 0.337 0.347
0.119 0.160 0.200 0.081 0.238 0.003 0.350 0.240 0.313
0.118 0.210 0.100 0.138 0.137 0.072 0.040 0.131 0.243
Large domains Lattice( 100) Search(lO0) Lattice(ll0) Search(ll0)
0.062 0.113 0.222 0.196
0.097 0.125 0.296 0.177
0.118 0.127 0.436 0.173
0.106 0.189 0.308 0.191
0.104 0.231 0.331 0.103
BERG
COHES:
AND
and a, [[llo]]
vectors,
ATOMIC
ARRANGENEKTS
and were required
IN
t’o contain
PARTIALLY
fraction
are in antiphase
more than three atoms and more than one branching
when the
at’om to
compared;
they
The items
counted
distinguish
“combined”
such
atoms
are
and t(wo-dimensional in Table
3 were
regions
those
The
intended
to
sensitive
to both shape and size of antiphase in Table
two dimensional of such
a
possible, e.g. (i.e. two
along
masks
4-atom
mask
along
[[lOO]].
was required t’o contain of antiphase atom
Pt
plane
as
regions,
as square
included
[[llo]]
indicated
masks
three
directions
were
in the scanned
as
i
(2) (111)
No
result’s
in Tables
2 and
3 clearly
between
the models introduced
syntheses
and the
simulations
adds
additional
literature as these
support
that’ tests
previous ones.
such
to
are
st’ringent
than
This is because ordered
TABLE 4. (8) Distribution
Run ::
already
simulations
are more
at,oms are in the
that
runs).
less than
cases
of misplaced
meaningful,
those
misplaced
of the
in
larger
of
are present
with
rat’io of the
cent, are
more
than
fractions
of
than those in [[llo]]. result in many
more
atoms
randomly
arranged,
are present
in [[lOO]]
regions
of misplaced
of 3 atoms
for all obher tesbs,
atoms
or more
except
platelem
because
or combined
[[llo]]
rods
or
combined
than
the
is
atoms w 3, atoms
in the other
can be distinguished rat,io of the
tesm.
fraction
in [[lOO]] rods to
and in
t#he fraction [[lOO]]
of
rods
Notme, however,
syntheses:
Frection Rods
[t(1oo)ll
CoPt, S = 0.80, qlmnnfrom experiment of misplaced
t[( 1lo)11
0.185 0.167
Pt atoms m 2-D regions >3 atoms >7 atoms 0.155 0.227
0.035 0.047
Run
[[(loo)11
1 2
0.495 0.379
Run
6 etoms (mm. of 5)
1 2
0.049 0.089
0.069 0.092
of combmrrl
[[(llo)ll 0.093 O.lOti
Pt atoms meet,ing the requirements
4 atomb (mm. of 4)
(d) Fraction
Pt atoms
Rods
of total misplaced
misplaced
0.075 0.090
Run
6 atoms (min of 5)
1 2
0.132 0.200
0.185 0.209
2-D regions >3 atoms >7 atoms 0.413 0.515
0.202 0.204
of the (110) masks
Mask size 9 etoms (mm. of 6) 0.070 0.094
Pt atoms meeting the requuvments
4 atoms (mm. of 4)
is
that
* 810 IS the total number of misplaced Pt atoms in this run. (b) Distribution of combmed Pt atoms
(c) Fraction
is
for the
the fract,ions of atoms in either kind of rod is smaller
Pt atoms in computer
0.470 0.499
fract’ions
20 per
arrays
of atoms
and (110)
other
most
Number of combined atoms/810*
381 404
the
and only a small
of misplaced
Total so of combined atoms
from
most of the Co and Pt
matrix
atoms
rods, and the fraction
(4) (100)
This in
of
(111) platelets.
in Lattice
(Search
details
rods than in [[lOO]] rods.
in two-dimensional
indicate
have
Over
the
all misplaced
and [[llo]] much
The
Also.
only a small number
the
1) are
for all the test,s.)
combined
arrays.
anti-ph ase platelem
(3) with
output. agreement
each.
atoms in [[llo]]
(110)
across
of the
atoms in [[lOO]] rods is greater
number
is evident
the following
in two-dimensional
atoms
and
table.
(This
qIrn,, in Fig.
similar
revealed
1585
8-I
atoms in rods of 0.2 or less, while at least
quarters
present
a region
once on a given
from
are very
in two-dimensional
2 x 9 Co
To be counted,
more than
these
were
regions.
al,,,,, (calculated
combined
rods from
at least the minimum
atoms
was counted
are
the size or compactness
These atoms
sites
t,wo at,oms
2 distinguishes
regions-not
regions.
masks
CoPt
the atomic configurations : (1) small 3-D antiphase regions
compact
of
data
The
rods
described
distinguish
atoms.
The
all
masks
clusters while the
Pt
rods.
comprising
regions.
misplaced
from
ORDERED
16 atoms (min. of 8)
0.064 0.121
0.094 0.126
of the (110) masks
Mask s,ze 9 atoms (min. of 6) 0.188 0.212
12 8t,oms (mm. of 7)
12 atoms (mm. of 7) 0.172 0.273
16 atoms (min. of 8) 0.251 0.284
ACTA
1586
for (110) the
platelets
fraction
in
than two
for (100)
METALLURGICA,
platelets,
dimensional
but that
regions
greater
than three atoms is larger. (5)
the
masks
are
useful
in
distinguishing
3-D
VOL.
APPENDIX. SEPARATION OF 18~0 AND INTENSITY DUE TO ATOMIC DISPLACEMENTS
plane from those on another. In
Table
4 the
m.easured
some
results are
q’s
indication
of two
given.
of the
errors
size of the model, 4000 atoms. test results state
in Tables
of long-range
misplaced
simulations
The
involved
give
with
Comparison
the
with the
2 and 3 shows that in a partial order CoPt,
contains
ant,iphase with the matrix. Further
If up to square terms in displacements
of atoms from
lattice sit,es are included, t,hen the fatal diffuse intensity at a point, uritten’21~22)
h,, h,.&
in reciprocal
space,
can
be
most of the
atoms in the form of small (100)
as for C&Au.
with
differences
1973
21. C’. J. SPARKSand B. BORIE. Local dtornic Arranaeww,tt.~ Studies by X-ray Diffractiorb, p. 53. edited by J. B.“C~HE\ and J. E. HILLIARD. Gordon & Breach (1966); .4&r tryst. A27, 198 (1971). 22. J. E. GRAGC, JR. and J. B. COHES, dcta Net. 19, 507 (1971).
regions or random arrays from platelets. They are not very useful in distinguishing platelets on one
the
21,
platelets,
This is the same situation
discussion
will be postponed
ISRO
=
(hl,h,2, h3)
+
h&(hl.
h,.h,)
until the results in Part II are presented. ACKNOWLEDGEMENTS
This
research
was
Science Foundation.
sponsored
by
One of the authors
fully acknowledges
an NDEA
western University
Cabell fellowship.
the
+
~2QX(~,~~,~
h3)
+
h2%&l,
+
~&2~~,(~23~2.
+
h2.4)
u?_y(~,.h1.h,)
i-
hz2B_y(h2.h,.
h,)
National
(H. B.) grate-
fellowship and a NorthPortions
work were submitted
by H. Berg in partial
of the requirements western University.
for the Ph.D.
degree
of this
fulfillment at North-
The
Fourier
REFERENCES
f
h~,~,,(~,~
coefficients
4.
of the
detailed
they
vanish
equations
here (however,
see Refs.
W)
involving
pract,ically be it is necessary
at, the eighth
for them
h?).
terms
the displacements (Q, R, S) cannot obtained in linear analyses (because to assume
1. P. C. CLAPP,Critical Phenomena in Alloys, Magnets and Superconductors, p. 299. edited by R. E. MILLS. E. &HER and R. I. JAFFEE. McGraw-Hill (1971). 2. J. E. GRAOGI,JR., P. BARDHA~ and J. B. COHES, ibid, p. 309. 3. H. N. SOUTHWORTHand B. RALPH, Phil. ,Uag. 14, 383 (1966): 21,545 (1970). Appl. Phys. Lett. 9, 7 4. T. T. TSON~ and E. W. MiiLmR. (1966); J. appl. Phys. 28, 3531 (1967). _ 0. A. H. GEISLERand D. C. I~IARTI~,J. appl. Phys. 22, 375 (1952). 6. El. BERN and J. B. COHEN, Afet. Trawls. 3, 1681 (1972). (For actual Bragg intensities of the crystal employed in this study with S = 0.82 see H. BERG, Ph.D. thesis, Sorthwestern Umversity, 1972.) 7. D. R. CHIPMAN,J. appl. Phys. 27, 739 (1956). 8. L. H. SCHWARTZ and J. B. C’OHES,J. appl. Phys. 38, 598 (1965). 9. T. ERICSSOS,S. LI~DE and J. B. COHES, J. appl. Cryst. 4, 31 (1971). 10. E. GEBHARDTand W. KOSTER, 2. &fetaZZfi. 32,253 (1940). 11. P. M. DE WOLFF, Acla Cyst. 9, 682 (1956). 12. L. H. SCHWARTZ,L. A. MORRISOS and J. B. COHEN,ddv. X-ray Analyeia 7, 281 (1964). 13. J. E. GRAOQ, JR. Ph.D. thesis, Northwestern University, 1970. 14. T+LRICHESSON,L. MORRISONand Ii. PAAVOLA, J. appl. Cry&. 4, 524 (1971). 16. B. W. BATTERMAN,D. R. CHIPMAXand J. J. DEMARCO. Phys. Rev. 122, 68 (1961). M. J. COOPER,Acta Crysl. 10,1067 (1963). ;:: D. T. CROMERand D. T. MANN, LASL Report LA-3689 (1967). D. T. CROMER,J. Chem. Phye. 50,4857 (1964). ::: P. C. GERLEN and J. B. ~OHES, J. appl. Phys. 40, 5193 (1969). 20. P. C. GEHLEN, Ph.D. thesis, Northwestern University, 1966.
h)
shell),
so bhe
n-ill not, be presented
21 and 22).
It is sufficient
to point out that Q is a sum over In/)/of sine functions, R and S involve
whereas
cosine
functions.
Q is a
series in terms of mean static displacements atoms from the average lattice sites.
of
Qs involves the sum over 1~01 of the X components of t’he displacements atomic
vector
of atoms separat’ed
lmn.
Since
the
by an inter-
mat,erial
is
cubic,
Qs (hlh2hs) = Q, (h2hlh3) = Q, (&h&J. because the displacements (X,,,) = ( Ynrln) = ::Xn,,llJ. Thus, 0111) Qs is required in equation Al : Q, and Q, do not appear, and similarly square
(static
includes
cross
for the t,erms R, S. and
d_ynamic)
products
of
R involves
displacements displacements
meanand
such
S as
(XrmnYlmn). Each of these terms has the symmetry shown in Table Al, which results in a minimum repeat volume in reciprocal must be obtained space.
Because
variations
space over which the component
to know
it throughout,
of these symmetries,
of the terms in equation
minimum volume required smaller than a unit cell.
for
reciprocal
and the different (Al)
with h, the
measurement
The separation of t’he terms along the three of interest will now be examined. Consider that term
is lines first
Isno (h,, 0,2) = Isno (h,, 0, 0), because this has a periodicity of 2 in reciprocal space.
BERG
ASD
COHEN:
ATOMIC
ARRANGEMENTS
IN
PARTIALLY
ORDERED
CoPt,-I
1X7
Al. Symmetries of the components of the total measured diffuse intenslt,y, for an f.c.c. material. All components hare a period of two in h,.
TABLE
Symmetry
Term h, = 0, h, =
ISRO
&$
(i =
h, + h, = 1, h, -
across
1, 2, 3),
h, = 1, h, = -h,
h, = 0, h, = 0, h, = *, h, = 4, h, = i-h,, h, * k, = 1,
Q,
h1 = 0, h, = 1 (whole line is antisymmetric) h, = 0, h, = +&, (i = 1, 2, 3),
x, S”.
11, = kh,,h,
& hs = 1
h, = 0, A, =
+&, h, = iha,
h, 3~ h, = 1,
h, = 0, h, rf 4, h3 = 0, h, = 54 h, = 0, h, = h, (i, j = 1, 2, 3;
IT
(n-hole line is antisymmetric)
i = j)
AeAilT(hl, 0, 2) = I,@,.
Then : I&
0,54 = Isa&,.
(42) + w,(~,,o~
+ w&,
h,* 0) + h?R&,
-+ 4R,(2, h,. 0) + 2V,,(2,
0>2)
0,2) - I,(2
-
Employing
= line I - line 2 = 2Q!,(h,, 0,2) + 4% -
l)JUh,, 092) (A2b)
+ 4&!,(2: A,, 01, A&@,:
0,2) = I,(2
- h,. 0$2)
Jr I,(2
+ IL,.0.2) -
21&L,, 0,2)
= line 2 i_ line 3 - 2(line 1) = SR,(h,, 0,2).
(A2c)
The lines are shown in the minimum repeat volume in reciprocal space in Fig. Al.
the
I,(2
-
h,, 0,2)
symmetries
(A2d) in Table
Al,
S,,
J2J
= I,(O. 0, 2 :
-
(h,, h,, h3) = -S,, t---h,, h,, &); therefore S,, (2, I+$) must be zero for all h,. Similarly S,, (h,, 0, 2) = S,, (2, h,, 0) = 0, and Q, (2, h,, 0) = 0. However, because there are errors in the actual separation due to statistical errors in any experiment these terms have therefore been included. All terms in equation A2a except Q, (2, h,, 0) and R, (2, h,, 0) are obtained from oquation (A2b-A2d.) Keeping in mind the symmetry in Table Al and the periodicity of two, it can be seen that R, (2, h,, 0) = R, (0, 0,2 - h) and Q, (2, h,, 0) = Q, (0, 0, 2 - IL,). Therefore, the remaining terms can be obt,ained from the following differences : A,I&O, 0, 2 -
“1
$2)
= PAS.&, 0, 2).
(A2a) h,, 0,2)
I&,,
- k,, 2,2)
= line 1 + line S - line 7 - line 2
A,, 0)
i ~~(O)~_~~(~~, 0, 2)* A1.z&,, 0,2) = I&,.
-
2)
0,2) + I,(2
h,) -
I&,
0,2 -
h,)
= line 4 - line 5 = “Q,(O, 0, 2 - n,) - 4(O)R,(O, 0,2 :
+ 2f2 - h&Y,,{2 i
A&(0.0,
2-
-
= 2 (line 5) = 8R,(O, 0,2 -
’ “2
(A24
n,)
= 21,(2.0,2
Fm. Al. The mmimum repeat volume in reciprocal space to separate mt,ensity due to focal order from intensity due to atomic displacements m an f.c.c. alloy. The numbered linear segments permit the separation along the h,, 0, 2 lines.
h,, O,O),
h,)
h,) -
21,(0,0,
2-
h,)
2 (line 4) h,).
Wf)
All size terms can therefore be evaluat,ed from measurements along eight. lines, three of which overlap, in a compact region of reciprocal space. [It is this attempt at compactness and to avoid the origin of reciprocal space that has led to the choice of the volume and hence the lines involved.] With
ACTA
1588
~ETALLURGI~A,
the size terms and equation (A2a,) Isno (hO0) can be evaluated. A similar analysis can be employed to determine ISno (h, h, 0), which is equal to lsRo (h,, 0, 2 - h,). All terms but S,, (2 - h,, n,, 0) are determined from A,l&h,, 0, 2 - h,) and from A,*fr(h,, 0, 2 - h,). The term &‘,,(I&,,h,, 2) = -S&2 - h,, h,, 0) and the former can be obtained, for h, = 0.0 - 0.5, from
VOL. 21, 1973 synthesized on a computer. (The required equat,ions for the size terms are in Ref. 21, 22.) At,omic displacements in Q, R. S were calculated for first TABLE A2. Comparison of the short,-range order parameters obtained from the trilinear analysis wzth the oragmai Input values
From analysis of synthesized data
Input
= I&h,, h,, 2) + I,(2 - I@,. -
I,(2
2 -
-
h,, 2 - h,, 2)
h,, 2) - I&h,, 2 -
h,, 2)
h,, h,, 2)
= as,, h,, A,, 2), end for h, = 0.5 &41&l, = 1241 - I,(1
(Asa)
-
h,, 1) + I,(1
+ hr. 1 + h,, 1)
h,, 1 + h,, 1) - I,(1
= PS(,,,h,% 4,2).
(from Ref. 8)
0 1 2 3 4
0.360 -0.037 0.034 0.001 0.007
5
-0.008
O.fOOi -0.009
0.000 - 0.002 0.012
- 0.003 0.000 0.015
6 7 8
1.0:
&2) h,, 1 -
i
+ h,, 1 -
h,, 1) (A3b)
Finally, consider the h,, h,, 2 - h, line, equivalent to h, h, h and extending from 0, 0, 2 to 0.5, 0.5, 1.5. Forming AJ, rend A,21, allows the determination of all needed quantities, when the symmetries in Table Al are considered. One of the major reasons for the attempt to make the measuring region as compact as possible (and for including the S,, terms along [hOO]) is that the size terms include scattering factor ratios. OnIy if these ratios are con&ant in the measuring region are the periodicities and symmetries strictly maintained. To test this trilinear procedure, data was
0.360 - 0.04 1 0.03ti ct.002
neighbors from the sizes of the atoms and were allowed to decrease, following the inverse-square law, with increasing interatomic distance. The scattering variation was allowed. Compton modified scattering was included as well as a. variation in surface roughness. The local order coefficients employed were those measured from Cu,Au. S = 0.8O.(n’) A direct beam was chosen to give 2000 counts per point. (Wit’h the monochromat,ic Co& radiation employed in this experiment’ the actual counting time would be 20 min per point.) A normally distributed random error was then added t,o the intensity at each “measured” point. The initial values of ?i and these values aft‘er analysis of this simulated intensity are shown in Table A2. The method appears adequate.