TEMPERATURE DEPENDENCE OF STACKING FAULT ENERGY by Craig M. Shevlin
TEMPERATURE DEPENDENCE OF STACKING FAULT ENERGY
IN A Cu-30w/o Zn ALLOY by
Craig M. Shevlin
A Thesis Submitted to the Faculty of the
DEPARTMENT OF METALLURGICAL ENGINEERING
In Partial Fulfillment of the Requirements
For the-Degree of
MASTER OF SCIENCE
WITH A MAJOR IN METALLURGY
THE UNIVERSITY OF ARIZONA .
1 9 6 9
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of
requirements for an advanced degree at The University of Arizona and
is deposited in the University Library to be made available to borrowers
under rules of the Library,
Brief quotations from this thesis are allowable without special
permission, provided that accurate acknowledgment of source is made.
Requests for permission for extended quotation from or reproduction of
this manuscript in whole or in part may be granted by the head of the
major department.or the Dean of the Graduate College when in his
judgment the proposed use of the material is in the interests of
scholarship. In all other instances, however, permission must be
obtained from the author.
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on. the'date shown below:
T7lT. J. DEMEE
Professor of Metallurgical Engineering
The author wishes to express appreciation to his advisor, Dr.
L. Jo Demer, for his help and interest throughout this study. His
guidance and understanding has made this work possible
Thanks is also extended to the National Science Foundation for
funds which made this project possible.
Andrew F , Chaloka
TABLE OF CONTENTS
LIST OF ILLUSTRATIONS
LIST OF TABLES . . . . .
' . . , , , ..........
I, I N T R O D U C T I O N
. . . .
. . . . . .. . . ,
. .. . . . o o
0 0 0 0 , 0
II. THEORETICAL BACKGROUND . . , , , ’, , , , , , , , , o ,
2ol Stacking Faults in Face-Centered Cubic Metals . , ,
2.2 Experimental Determination of Stacking Fault
Energy . , . , , , , , , , , , . . , , , , 0 o , o
2,3 Stacking Faults Studied by X-Ray .Diffraction . , » .
2o3©l Calculation of Faulting Parameters . , , , ,
2,3*2 Calculation of Stacking Fault Energy , , , ,
2o4 Temperature Dependence of Stacking Fault Energy «
III. OBJECTIVES o ................
4ol Specimen Preparation .
0 0 0 0 0 0 , 0 0 0 0 0 0 0 0
4 ol ,1 Melting and Casting . , , , , , , , o , , , ,
4olo2 Chemical Analysis of . » . o » , ,
4 ol o3 Preparation of Brass Filings . ..........
4.2 X-Ray S t u d i e s ................. . , , . . , , . . .
4,2,1 Depressed Temperature Studies , , , , , , , ,
4 o2o2 Elevated Temperature Studies. , , , , , , , ,
4.3 Analysis of Data . , , , . , , , , . , , , , , , , ,
• o , o , 4 » o , , o o ,
5*1 Low Temperature Results
5*2 Room Temperature
• . • • • • , , ■ ■ 57
• „ , , « , 59
5*3 Elevated Temperature • o , • ,
5*4. Discussion of Results , , o o • « , .......... ..
68 VI, CONCLUSIONS . . . . . .................. , , , , , , , ,
V I I . SUGGESTIONS FOR FURTHER WORK . , .....................
SELECTED BIBLIOGRAPHY . , ....................... . . . . . .
Page vi viii ix
57 r ' .
v / 4 . \ /.
LIST OF -ILLUSTRATIONS
1- Dissociation of a unit edge dislocation into partials.
in the FCC structure . . . .......... .
2- Dissociation of a unit screw dislocation into partials in the FCC structure • . • «
3o Dissociation of a dislocation of mixed orientation into partials in the FCC structure „ « « « « o o « o o »
4. Surface tension of faulted area balanced by repulsion force, between partial dislocations . „ „ „ „ „ 11 in the FCC structure
6, Stacking associated with an extrinsic fault in the
« e „ o
?o Stacking associated with a twin (growth) fault in the
FCC structure . o . o o o . o o. .o' «
8 e Tetrahedron describing Thompson' s notation for dislocations in the FCC structure « e . .
9o Unit dislocation node with Thompson's notation . „ . . .. .
1 0 o Extended node and stacking fault formed by partial dislocations « « « © © © © © © © © © © © < © © © © © © ©
11. Contracted node and stacking fault formed by partial dislocation • © © . . « © „ . . . . . • © . © . © © © © ©
12© Typical In A versus h curves for first two orders of (111)
......... . . . . . . . . . . . . .
PF versus L curve for determining effective
13 l4 l6
14." Typical a ^ ^ v versus cos© cot© curve for cold-worked filings © © © © © © © © © © © © © © © © © © © © © © © © ©
13© Free energy as a function of temperature for cobalt © © © ©
. . . V I
LIST OF ILLUSTRATIONS— Continued
. Sample holder for x-ray studies . . . . . . . . . . . . . .
17. Sample holder in loading position
18. Sample holder in position on diffractometer
19. Comparison of (ill) profiles for
° K and 373° K
20. Comparison of (ill) profiles for 100° K and 300° K „ . . „
LIST OF TABLES
1 o Detailed Chemical Analysis of High Purity Copper
and Z i n c .............. . . „ ........ ..
Diffraction Analysis . . • „ -o!
The stacking' fault energy of Cu-30w/o Zn filings deformed at
liquid nitrogen temperature was determined as a function of temperature«
The x-ray diffraction method was used. Measured values of the stacking
fault probability and mean squared strains were used to calculate the
stacking fault energy. It was found that the stacking fault energy of
this alloy does not change significantly when the temperature is raised
from 100° K to 300° K. The measured stacking fault energy agreed
closely with published room temperature electron microscopy -results.
Analysis of the filings at 373° K gave anomolous results.
I . INTRODUCTION
An important area in the field of physical metallurgy is the
study of stacking faults in metals and alloys. In recent years emphasis
on studies in this area has increased as the effect of stacking faults
on the plastic deformation characteristics of metals has become known.
Stacking faults affect not only the movement and distribution of dis
locations during plastic deformation but also have a strong influence
upon electrical properties (Christian and Spreadborough, 1956), texture
(Valenzuela, 1965 ), recovery and recrystallization (Bailey and Hirsch, ' i
), and other phenomena as well.
Face-centered cubic (FCC) and hexagonal close-packed (HCP)
crystal structures both represent the closest possible packing of atoms„
In the FCC structure close-packed layers of atoms are stacked one above
the other in such a way that the atoms in the fourth layer are directly
above those in the first layer. This results in the stacking sequence
ABCADCA where A, B, and C represent the positions of the atoms in the
respective layers. In the HCP structure the close-packed layers are
stacked so that the atoms in the third layer are directly above those in
the firsto This results in the stacking sequence ABABABA. A stacking
fault occurs when there is a region of HCP’stacking in a FCC structure
or a region of FCC stacking in the HCP structure.
Stacking faults are produced in metals during crystal growth,
plastic deformation, and phase transformations as well a s •through the.
collapse of vacancy disks. The faulted areas are usually considered to
: - 1 .
■■ ■ ■
be analogous to ribbons with widths on the order of 200 A for faults
produced by deformation in the alpha brasses (Warren, 196l). The
present study is mainly concerned with faults produced by deformation in
the FOG structureo
All stacking faults have a finite energy associated with them
called the stacking fault energy. Christian and Swann (
) define the
stacking fault energy as the change in free energy of a crystal per unit
area of fault introduced. As will be shown in a later discussion the
stacking fault ribbons are actually bounded by partial dislocations
supplies a constant attractive force. When this attractive force is
sufficient to balance the elastic repulsion of the partials the total
energy is minimized, and an equilibrium ribbon width is attained.
it is clear then, that in a metal with a high, stacking fault
energy such as aluminum or nickel the equilibrium width of the ribbons
will be quite narrow, since only a small separation is required to
balance the repulsive force exerted by the bounding dislocations. In
metals with a low stacking .fault energy such as some copper and silver
alloys the equilibrium ribbon width is much larger so that the total
energy of the faulted area will be sufficient to balance the repulsive
force of the partial dislocations.
The significance of the stacking fault energy as a contributing
factor in plastic deformation becomes evident. The stacking fault
ribbons must come together before a dislocation can leave its slip plane
and move on an intersecting slip plane. It follows then that this cross
slip is favored by a high stacking fault energy since the narrow ribbons
are more easily forced together by an applied stress e It is for this
reason that slip bands observed in aluminum appear wavy while those in
an alpha brass with its lower stacking fault energy are straight. This
is only one of many observed effects in which the stacking fault energy
plays an important role.
.In many early treatments of the behavior of FCC metals the
stacking fault energy was treated as a constant and was not considered
to. vary with temperature-. In Seeger's model (Diehl, Mader, and Seeger,
) cross slip takes place when partial dislocations are forced
together over a critical portion of their length and the dislocation
expands on the cross slip plane. The stress required to recombine the
partials is considered to be supplied by dislocation pile-ups and
thermal fluctuations. No effect of temperature on the stacking fault
energy is considered. If the stacking fault energy were temperature
dependent, however, it would be expected that the partials would be
forced together, by an increase in the stacking fault energy with
increasing temperature, More recently, experimental evidence has shown
that in certain alloy systems there is indeed a temperature dependence
of stacking fault energy, Using the transmission electron microscope,
Swann and Nutting (i
) observed the collapse of stacking fault ribbons
in Cu-r7w/oAl upon heating, This indicates an increase in the stacking • fault energy with temperature. Using the node method of calculating t h e .
stacking fault energy, Ericsson (
) found an increase in stacking
fault energy in some Co-Ni alloys with increasing temperature,
The objective of the present study was to determine if there is
a temperature dependence of stacking fault energy in an alloy of cold
worked Cu~30w/oZn. The x-ray diffraction method was used*
This study is one of a series of investigations into the nature
of stacking faults which have been carried out in the Department of
Metallurgical Engineering at The University of Arizona. Valenzuela•
) studied the effect of the stacking fault energy on the texture
transition in the alpha brasses. Cotter (
) studied the stacking
fault probability of aluminum alloys of silicon and magnesium as a
function of composition. Simmons (
) determined faulting proba
bilities in eutectoid steel wire in an effort to correlate this with
fracture characteristics„ Hinton (
) determined stacking fault
probabilities in electrodeposited copper.
II. THEORETICAL BACKGROUND
2.1 Stacking Faults in Face-Centered Cubic Metals
The planar surface of separation between two regions of a
crystal which have the same orientation but which do not form a continu
ous lattice is called a stacking fault. In the FCC structure there are
three types of stacking faults which can occur. They are the intrinsic
fault (deformation fault), the extrinsic fault (double deformation
fault), and the twin or growth fault.
The normal FCC stacking is given as ABCABCA. It may be
expressed by the Frank (l$)5l) notation as A A A A A A where each A
represents a transition of the type A -> B -> C -* A. HCP stacking is
of the type B -* A -» C -» B. The stacking associated with an intrinsic
stacking fault in the FCC structure is ABCAB/ABCAB or A A A A V A A A A *
The fault is indicated by the slant bar and is a small region of HCP
stacking. This type of fault is also known as a 1 A fault. It is
equivalent to the removal of a portion of one atomic Flllj plane from
the crystal, although it is not formed in this way as will be shown.
This type of fault is by far the most important in FCC metals and is the
main concern of this study.
The stacking associated with the extrinsic or double deformation
to the removal of a portion of two atomic ^lll^j planes or the insertion
of a portion of one. This type is also called a 2 A fault. The growth
twin fault is the third type encountered in the FCC structure. The
stacking pattern here is ABCAB/ACBAC or A A A A W V V V with the lattice
beyond the fault being the twin of the parent lattice. Twin faults form
during plastic deformation as well as in the processes of solidification
and recrystallization. It should be noted that the insertion of an
intrinsic fault at every (ill) plane of a FCC structure gives the (ill)
twin of that structure, while an intrinsic fault introduced at every
other plane gives the HCP structure. While more complex fault struc
tures are encountered in the FCC structure they are best regarded as
combinations of these basic types. That these faults are the result of
certain dislocation interactions will now be shown.
A dislocation whose Burgers vector (b) is a unit lattice
distance is called a "complete," "unit," or "perfect" dislocation. When
such a dislocation glides on a j\llj plane it leaves the crystal struc
ture unchanged. Such a unit slip dislocation in the FCC structure is
specified by giving the magnitude and direction of its Burgers vector.
Thus a/2 [lie] specifies a dislocation whose Burgers vector is one-half
of an identity distance in the  direction. Such perfect disloca
tions are not commonly found, however, in FCC metals and alloys. More
often it is found that these perfect dislocations undergo a splitting
process in which two "partial" dislocations are formed. A typical
reaction is: a/2 [lie] = a/6[l2l] + a/6[21l]. The two product disloca
tions are the partials and each has a Burgers vector smaller than the
perfect dislocation. This reaction is allowed to occur due to a reduc-
tion in the total strain energy which is proportional to b (Frank,
Dislocations of the type a/2<^.l^> are called Shockley, partial
dislocations and are glissile since the Burgers vector is b in the slip
planeo Figure la shows schematically the dissociation of a unit edge
dislocation lying in a j^lllj plane . The resulting partials .are of mixed
of the partials are of opposite sign and attract each other. The edge
components, however, are of like sign and being larger in magnitude,
cause a net repulsive force to exist between the partials. This causes
the partials. to separate in the slip plane. The area separating the
partials is a region of intrinsic stacking fault. The entire arrange ment of the two partials and the stacking fault is an extended disloca tion. The dissociation of dislocations of screw and of mixed orienta
tions are shown in.Figs. 2 and 3, respectively. In each of these cases
the Shockley partials separate in the slip plane until the repulsive
force they exert on each other is balanced by the attraction caused by
the energy of the faulted area. Figure 4 shows this relationship in
graphical form. As the width of separation increases the repulsive
force between the partials decreases and the total faulted area
increases. At some width w the total energy of the faulted area is
sufficient to balance the repulsive force. This is the equilibrium
separation. In a metal with a low stacking fault energy the partials
can separate to a greater width than in a metal with a high stacking
A dislocation with its Burgers vector non-parallel to the fault
plane is sessile and can move only by climb..- The Frank partial is such
a dislocation and its Burgers vector is a/3^Llh> . It can be formed by
> b unit
a. Dissociation of a unit edge dislocation lying in a ^112^ direction
into two partial dislocations.
<n§> b' screw b” screw b* edge b" edge
b. Vector form of (a) showing the repulsion of the edge components of
Fig. 1. Dissociation of a unit edge dislocation into partials in the
> b unit
Dissociation of a unit screw dislocation lying in a ^112y
into two partial dislocations.
<L12> b' edge b* screw
b" screw b” edge
b. Vector form of (a) illustrating the repulsion of the screw
components of the partials.
. Dissociation of a unit screw dislocation into partials in the
a. Dissociation of a unit mixed dislocation lying in a ^ 110^
direction into two partial dislocations.
b' edge screw b” edge
b. Vector form of (a) showing the repulsion of the edge components of
F i g . 3. Dissociation of a dislocation of mixed orientation into partials
in the FCC structure.
Fig. 4. Surface tension of faulted area balanced by repulsion force
between partial dislocations.
the removal or insertion of a portion of a close-packed plane of atoms.
This could take place by the coalescence of a platelet of vacancies or
interstitials. The former case is shown schematically in F i g . 5 and the
stacking fault that has formed is an intrinsic one. If a close-packed plane is inserted as in F i g . 6, the dislocation loop is still of the
Frank type but the stacking fault is an extrinsic or 2 a
one. The Frank
dislocation has nearly twice the energy of a Shockley dislocation.
Friedel (1964) illustrates the greater distortion of the crystal in the
case of a Frank partial. This energy can be reduced, however, by the
splitting of a Frank dislocation into a Shockley partial and a stair-rod
dislocation. Figure 7 shows the stacking associated with a growth twin.
as thick as an intrinsic stacking fault.
Thompson (1953) has described a convenient notation for disloca
tions in the FCC structure. A tetrahedron such as the one shown in
F i g . 8 is utilized. The tetrahedron has corners AECD and faces a, ^ ,
^6 , and
corresponding to the four sets of |lllj planes in a FCC
crystal. Burgers vectors of unit dislocations are represented by the
edges of the tetrahedron while those of Shockley partials are repre sented by lines such as A<5", B
represents the Burgers vector of a Frank dislocation
and is normal to a j\ll| plane. This notation best illustrates the
vector properties of dislocations and gives a clear picture of the
geometry of some dislocation reactions. Thus the reaction BA = B(5 + <5A
is an example of the dissociation of a perfect dislocation into two
Shockley partials with fault layer between them. The familiar
3 ' a
. Stacking associated with an intrinsic stacking fault in the
Fig. 6. Stacking associated with an extrinsic fault in the FCC
. Stacking associated with a twin (growth) fault in the FCC
Fig. 8. Tetrahedron describing Thompson's notation for dislocations in
the FCC structure.
Lomer-Cottrell lock (Lomer, 1951? Cottrell, 1952) is formed by the
reaction D + C <5 + B = D /^ + >^<5 + <5 B. Here two dissociated dislocations on intersecting slip planes react at the line of inter
section to form a configuration consisting of two Shockley partials with
their stacking fault areas and a stair rod dislocation. This arrange
ment is thought to be a barrier to further slip thus contributing to
strain hardening. A dislocation node is formed when three unit disloca
tions meet as in F i g . 9* When three unit dislocations in the same plane
such as AB, BC, and CA split into the partials A
+ B, B
+ <5C, and
C (5 + <5A, a partial dislocation node may form. The partials then
assume one of the configurations shown in F igs. 10 and 11 where the
shaded portions represent areas of stacking fault.
An important parameter in the study of stacking faults is the
intrinsic stacking fault probability Cf. This quantity is really a
measure of the amount of faulting that is present. As it is a proba
bility value it can never exceed unity. A value of the stacking fault
indicates that out of one thousand^lllj planes,
there is a probability that fifty are completely faulted. A completely
faulted plane in a crystal is one whose entire area, to the boundaries
of the crystal, is covered by stacking faults. As Gf approaches unity,
a FCC structure becomes completely faulted and reverts to the HOP form.
Patterson (1952) has shown that when Of is small there is no difference
between a few planes completely faulted and many planes only partially
faulted. This is important in x-ray studies because the diffraction
effects are the same in either case. It is important to emphasize that
Fig. 9- Unit dislocation node with Thompson's notation.
Fi g . 10. Extended node and stacking fault formed
Fig. 11. Contracted node and stacking fault formed by partial disloca-
the stacking fault probability is not a fundamental quantity. For a
given metal it can vary widely with degree and type of deformation.
The reciprocal stacking fault probability, of ^ a quantity used
by Warren and Warekois (1955)i is also quite useful. It is defined as
the average number of unfaulted planes separating the faulted planes.
Thus a value of 20 for
^ corresponds to
equal to 0.05 and indi
cates that on the average every 20^ plane is faulted. The probabilities
of extrinsic faults and twin faults are designated as Qf and
The relationship between cf and the stacking fault energy
not as easily interpreted. A high value of
can correspond to either
a wide equilibrium spacing of partials or to a large number of closely
spaced faults. In the former case the stacking fault energy is low. In
the latter case it is high. In order to make assumptions about the
stacking fault energy from cf it is necessary to know the existing dis location density and configuration. For a constant dislocation con sidered to be proportional to Of ^ (Vassamillet and Massalski,
The energy of a stacking fault originates from the forcing of
electrons into higher energy states in the region of HCP stacking. A
clue to the stacking fault energy is obtained from studying the nearest
and next nearest neighbor relationships. Weertman and Weertman (1964)
point out that to a first approximation the stacking fault energy is
zero since an atom in a faulted region, either intrinsic or extrinsic,
has the same number of nearest neighbors as an atom in a perfect
lattice. The energy of the fault compared to an unfaulted structure
arises in the difference in the number of next nearest neighbors. Here
the extrinsic and intrinsic fault energies appear to be equal• When
next-next nearest neighbors are counted however, it is found that the
extrinsic fault has the higher e n e r g y S i n c e an intrinsic fault can be
considered to consist of two coherent twin faults lying on adjacent
planes, the energy of a twin fault should be about half that of an
intrinsic stacking fault (Fullman, 1951)• In fact this relationship has
been used to arrive at experimental values of stacking fault energy by a
method which will be described in a later section.
2-2 .Experimental Determination of Stacking Fault Energy
There are several methods of measuring the stacking fault
energies of metals and alloys, These will be described here briefly.
The x-ray method used in this study will then be described in more
detail0 Christian and Swann (
) give a complete review of each of
The node method of measuring stacking fault energies was first
described by Whelan (1959)« In this method the stacking fault energy
is determined from the curvature of partial dislocations at extended
■nodes in lightly deformed material. Although the node radius is easily
measured using, the transmission electron microscope, it is more diffi cult to use this information to calculate the stacking fault energy.
The equations have undergone continual refinement however, and this
method does seem to yield the most reliable values of stacking fault
The ribbon method utilizes the electron microscope to measure
the spacing between partial dislocations. If it is assumed that the •
equilibrium separation is determined only by the energy of the fault and
the elastic repulsion between the partials then the stacking fault
energy can be calculated. This method can only be used if the stacking
fault energy is sufficiently low that the spacing between the partials
can be measured. The method is described in detail by Amelinckx and
The twin boundary method has as its basis the relationship between the twin boundary energy and the stacking fault energy
The stacking fault energy is considered to be equal to approximately
twice y. since an intrinsic stacking fault is really equivalent to two
coherent twin, boundaries on adjacent planes. The ratio of the twin
boundary energy to the grain boundary energy can be determined by
measuring angles of intersection between twin boundaries and grain
boundaries. Since the grain boundary energy can be determined inde pendently (Inman and Tipler,
), the twin boundary energy is known.
Unfortunately the relation
J = 2 J ^
is now thought to be incorrect
(Thornton, Mitchell, and Hirsch,
) and the values of stacking fault
energy from this method appear to be too low.
The onset of the third stage of plastic deformation in FCC
single crystals is thought to be associated with slip from the primary
slip plane onto the cross slip plane (Diehl et a l ., 1955)• The stress
T'lH necessary to bring on cross slip is dependent upon temperature and
strain rate. The stacking fault energy of FCC metals can be determined
by an analysis of the dependence of on the temperature and strain rate (Haasen, 1958). The stacking fault energies of pure metals
20 measured by the method are normally higher than the results from node measurements. It is not certain where this difference arises.
Stacking fault energy can also be determined by a method
requiring a knowledge of the free energy of a tilt boundary of dis sociated dislocations. Li and Chalmers ( 1963
) have used this method to
arrive at a value of stacking fault energy for pure copper of 25 erg/cm .
This value, however, is considered to be unreasonably low compared to
results from node measurements.
Friedel (1964) shows that it may be favorable for a Frank dis location to split into a stair red dislocation and a Shockley disloca tion by a reaction of the type:
A O' = A/3
+ Cf, using Thompson’s
notation. In a Frank loop this splitting often takes place on three cf
the ^ll6/» type directions and continues until the Shockley dislocations
meet and react to form stair rod dislocations along the intersections
of their glide planes. The result is a tetrahedron with stair rod dis
locations along its edges and stacking fault on its faces. These
stacking fault tetrahedrons are very stable. Czjzek, Seeger, and Mader
(1962) have developed a method for calculating the stacking fault energy
knowing the maximum stable size of a tetrahedron. It is difficult, how
ever, to determine if a tetrahedron has been able to reach its maximum
2.3 Stacking Faults Studied by X-Ray Diffraction
Plastic deformation of a FCC metal brings about several impor
tant changes in the internal structure. Among these are the formation
of inhomogeneous residual strains, small coherent domains and intrinsic
: : ■ '■. , stacking faults. Each of these has an effect on the x-ray diffraction
pattern of the metal» In order to effectively study stacking faults by means of x-rays it is important to consider all of the effects of cold work on the diffraction pattern.
When a metal is uniformly stressed it has its interplanar
spacings altered to new values. These changes can be measured by the
displacement of Debye rings in the powder diffraction pattern* In the
case of inhomogeneous residual strains resulting from residual micro-
stresses the interplanar spacings take on a range of values with the
result that a diffraction line is broadened into a corresponding range
of Bragg angles♦ The mean interplanar spacing is the same as in the
unstrained condition. The reciprocal lattice for such a non-uniformly
strained crystal is compressed where the crystal is stretched and
expanded where the crystal is compressed. If the stress varied in
magnitude but was constant in direction each reciprocal lattice point
would be stretched into a line. In any given reciprocal lattice row 1 the length of the lines would be proportional to the distance from the y
origin. This is true because points farthest from the origin correspond
to planes with the smallest d spacing. Thus a stress will cause d for these planes to be a larger percentage of d* Normally, of course, the
stresses vary in direction as well as magnitude and the reciprocal
lattice points elongate into bundles of non-parallel lines filling a
small volume. The sizes of these volumes is also proportional to the distance from the origin in any radial row. Since a radial row in
reciprocal space consists of the different orders of a particular (hkl)
reflection, the broadening of a diffraction line due to inhomogeneous
residual strains is dependent upon the order of the reflection. The
broadening is greater for higher order reflections.
During plastic deformation the size of the regions that diffract
as coherent crystallites is reduced. This is a consequence of the
general disruption of the lattice and the increase in dislocation
density. The particle sizes in cold-worked filings are often on the
order of 200 A (Warren, 1959)• The deformation has not formed discrete
particles of this size but has produced some type of domain structure
within the filings. When the size of the coherently diffracting domains
becomes less than about 1000 A the diffraction lines become broadened
symmetrically. The reason for this is that some of the rays scattered
at non-Bragg angles from the top portion of a particle do not undergo
destructive interference. The particle or domain is so small that the
planes which would diffract the cancelling rays simply do not exist.
The broadening of a diffraction line is described by the Scherrer (1920)
formula: t = kJ^/B cosOg. Where t is the particle size in cm., ^ is
the x-ray wavelength, B is the line broadening in radians at one-half
the peak maximum, k is a constant, and ©g is the Bragg angle.
In considering the reciprocal lattice for a very small crystal
it is found that the points become enlarged into volumes but that the
volumes are of equal size for points in any radial row. Thus A r * ,
where |r*j is the length of the reciprocal lattice vector, is constant
for different orders of h k l . Unlike strain broadening then, broadening
due to small particle size is the same for different orders of a reflec tion .
Intrinsic and extrinsic stacking faults and twin faults are also
produced during plastic deformation and produce changes in the x-ray
diffraction pattern. Wagner (1937) gives a complete treatment of these
effects. Much of the original theory was developed by Patterson (1952).
Wagner, Tetelman, and Otte (
) point out that the theory of diffrac tion effects from faulted structures rests on four assumptions:
1. The faulted layer extends across the entire coherently dif fracting domain.
2. Where the faulting plane is one of several variants (such as
|lllj in FCC crystals) faulting occurs on only one variant.
In the case of alloys the scattering power of all j\l]j
is the same.
4. Faults occur at random. There is no clustering of faults.
) found that assumption (l) is valid for a number of
FCC metals and alloys including alpha brass. It is doubtful whether
faulting in FCC metals is restricted to only one set of |"lll^j planes.
In cases where faults due to cold work are distributed on all sets of
j\llj planes Otte (1954) and Warren and Warekois (1955) show that the
effect on the x-ray pattern is simply due to the sum of the faulting
effects on all sets of planes. Thus
can be considered to be the sum
of the fault probabilities for the four sets of j\ll^j planes. Assump tion (
) is thought to be valid in the case of alloys deformed at low
temperatures (Wagner et a l ., 1962). There is evidence (Otte, 1957) that
faults do tend to form in clusters. However, Wilkens (i
) shows that
any non-randomness of faulting is not likely to produce any effect more
serious than a percentage error in the measured faulting probability.
In the FCC structure intrinsic stacking faults cause a broaden
ing and a shift of certain (hkl) reflections (Patterson, 1952). Those
reflections which have h + k + 1 = 3% where n is an integer are not
affected by stacking faults. Reflections with h + k + l = 3 n - l are
those that have had their structure factor, , altered by the
faulting. In the reciprocal lattice all points in a given class [i.e.,
(ill), (222), (333)] are elongated equally due to the faulting. The
broadening due to stacking faults then is the same for different orders
as is also the case with crystallite size broadening. The displacement
of a diffraction peak as a consequence of faulting can be used to calcu late the faulting probability Of as will be shown.
The effect of twin faults is to asymmetrically broaden a dif
fraction line and produce negligibly small peak shifts (Warren, 1959)•
There have been several measurements of the twin fault probability
from peak asymmetry measurements such as the displacement of the peak
maximum from the center of gravity. Willis (1959) has shown, however,
that the principal effect cf segregation in FCC alloys is to make
reflections asymmetrical. Wagner et a l . (19&2) point out that it is
difficult to separate the component of asymmetry resulting from twin
faults and that caused by segregation. Therefore the twin fault proba bility has not been reported in this study.
Extrinsic faults also cause a shift and broadening of diffrac tion lines. Warren (1959) has shown that the shift is equal in magni tude but opposite in direction to the shift caused by intrinsic faults.
The measured fault probability should therefore be equal to
Of - Of .
Wagner and Helion (
) conclude that there is no evidence of extrinsic
faults in alpha copper-zinc filings. Barrett and Massalski (
that extrinsic faults are rare but have been found by Otte and Chessin
) in cerium. Extrinsic faults are therefore not considered to be
of importance in this study and the faulting probability is taken to be
due to intrinsic faults only. The broadening contributed by extrinsic
faults is an asymmetric broadening unlike that caused by intrinsic
faults. In the absence of extrinsic faulting this is of course
2.5.1 Calculation of Faulting Parameters
The broadening of powder pattern peaks is a result of faulting,
small coherent domains, and micrcstrains. Wagner, Boisseau, and Aqua
) have shown that the profile of a diffraction peak can be repre sented in terms of a Fourier series:
where K is a constant given by Warren (1959) which depends on the
experimental arrangement. L is a distance normal to the reflecting
planes. © is from the intensity location © in P(29). ©^ is the posi tion of the observed peak maximum. and B^ are the Fourier cosine and sine coefficients, respectively. Since the functions dealt with here
are symmetrical functions, the sine coefficients are zero and we are
left with a Fourier cosine series:
[ c o s
47TL(sinG - sinG^)/^J .
Warren and Averbach (1952) show that the coefficients A^ which describe the broadening of the peak are really the product of two coefficients.
PF D PF
Thus A^ = A^ • A^ where A^ is the part related to broadening from small particle size (i.e., small coherent domains) and faulting. A^ is the part due to inhomogeneous residual micro-strains. Instrumental broadening is removed through the use of an annealed standard in which the only line broadening is due to instrumental errors. These would include any displacement of the sample from the diffractometer axis and the use of a flat sample rather than a curved one.- No broadening due to faults, lattice strains, or particle size exists in the annealed sample.
The Fourier coefficients of the annealed peak, A _ , > are used to
JAann) remove the instrumental broadening frcm the cold-worked sample using the
Stokes (1948) relation, A = A^, x/AT , x . A are the cold-worked
L TAcwJ JAann; L coefficients corrected for instrumental broadening.
Wagner and Helion (
) show that when the peak maximum is
taken as the origin of the Fourier transformation the coefficients A^
and A^ are given b y :
1____ fro* 0 > l-5(Cf + Of1 ) + /g v ao
where ) is the size of the coherently diffracting domains normal to the reflecting planes. T is the width of the domains in the |111 planes (i.e., the average width of the faulted region). cos0 is the average value of the cosine of the complement of the angle between the
^hk^> direction and the ^111> direction. Of, cf1 , and
are the probabilities previously discussed. aQ is the lattice parameter and V
2 2 2 2 is a constant given by Warren (1959)• h Q = h + k + 1 for the reflec tion being considered. is a component of the residual strain
which is normal to the reflecting planes. It is an average value over
the length L, squared, and averaged over all regions of the sample.
Warren (1959) points out that while this is only one component of the
strain it is often multiplied by three for use in computing a mean
strain energy. The relation is: V = 3E <|f^>/2 (Smallman and
Westmacott, 1957)• E is Young’s modulus.
Warren and Averbach (1952) have shown how to separate the cosine
PF D coefficients
into their two components A£ and A £ . From reciprocal
lattice considerations, as discussed previously, the broadening due to
the residual strains varies with the order of the reflection. The
quantity h Q in Eq. (4) shows this dependence upon the order. It was
also noted that the broadening due to small particle size and faulting
does not depend upon the order of the reflection. Equation (
this out. Writing the expression for A^ in logarithmic form:
,D and replacing A (strain term) by (4) gives
In = In
+ In p p p
1 - ZTT ITh
1967) shows the steps for the reduction of E q . (6) into workable
form using a Taylor series expansion. The final form of (6) is:
is then plotted versus h^ for multiple orders of (hkl). The
curves are plotted for values of L in 10 A increments. Typical curves
are shown in F i g . 12. Examination of Eq. (
7) shows that since A^ is
dependent upon the order of the reflection, extrapolation of these curves to h = 0 gives values of In A:
PF o --- L "
If these values are non-zero then part of the broadening must be due to particle size and faulting.
The first derivative of Eq. ( 7) is: d(ln d h
-2 7T2L 2
which upon rearrangement yields the mean squared strains knowing the
2 slope of the plot of In A^ versus h ^ . Thus:
d h 2
The knowledge of the mean squared strains is necessary to the final
calculation of the stacking fault energy.
Fig. 12. Typical In versus h Q curves for first two orders of (ill).
Warren and Averbach (1952) then plot the
PF values obtained
from the intercept at h " = 0 against their corresponding L values. This
is shown in Fig. 13. L is the previously mentioned lattice distance in
angstroms perpendicular to the reflecting planes. The intercept of this
line with the L axis gives a value for the effective particle size
normal to the reflecting planes. This is an effective particle size
since it includes the effect of faulting. If the true particle size is
desired this contribution due to faulting must be removed.
The density of stacking faults (stacking fault probability) can
be calculated directly from peak displacements in a cold-worked powder
sample. It is for this reason that cold-worked filings are normally
used in x-ray stacking fault studies. Wagner et a l . (
) point out
that cold-worked bulk samples contain residual elastic strains that may
not sum to zero, as well as possible preferred orientation. This net
elastic strain results in peak displacements independent of any faulting
effects. In randomly oriented filings however, residual macrostresses
sum to zero and peak displacements are due to faulting alone.
The peak positions from the x-ray pattern of a well-annealed
powder sample (e.g., without stacking faults) can be converted to
lattice parameter values. If these are plotted against cosG cotO, all
the values fall on a straight line. CosG cotO is an extrapola
tion function, f(0), suitable for the geometry of the diffractometer.
Extrapolation to cosG cotO = 0 yields the true lattice parameter. When
the material contains stacking faults the lattice parameters calculated
from the cold-worked peak positions, a (}1^ )'’ deviate in a systematic
way from the straight line. Four values of a (kkl)' are normally
Effective particle siz
Fig. 13• Typical A
versus L curve for determining effective particle
calculated using the positions of the first and second order (ill) and
(200) peaks. The theory of faulting in FCC metals (Warren, 1939) shows
that the deviation of the a ^ ^ , values from the straight line is
according to the values of the constant G. These values are given by
Wagner and Helion (
) and these values are: = -3*^3 x 10
G (200) = 6 '89 x 10"2 - G (222) = x 10"2 ’ G (400) = -5 -45 x 10'2 '
Since G (111) cy -1/2 G (2oo) and G (222) ^ "1^2 G (4 q o
) the straight line in
a plot cf a (hkl)’ versus cosG cot© can be drawn through the points a' = [ 2a(i:L1). + a (200)']
and a" = [2a(222)' + a (400) ’] ^ at
f e =
+ ) 0 / a d ) = f e +
respectively. As shown by Wagner et a l . (
) the stacking probability
G( can be calculated without the use of an standard from the equation:
S f r <10>
A a (hkl) = a (hkl) ’ ~ a (hkl)1 (ll)
A a (hki) Is thus the deviation of the lattice parameter a » (calcu
lated from the experimentally observed peak position) from the straight
line passing through the points a 1 and a". The point a(hkl)1
line is given by: a (hkl)1 =
a + m cos ©
where a is the extrapolated lattice parameter. The line given by
Eq. (12) and the points a0 are sllown in Fig. 14.
A value for
is calculated for each of the experimental!y measured
peak positions using Eq. (10). It is important to emphasize that the
theoretically derived constants ) make it possible to calculate Cf
from this method. Without them it would be necessary to calculate of
from peak position differences between an annealed standard and the
cold-worked sample. This requires that the extrapolated lattice
parameters for the annealed and cold-worked samples be identical. This
is a very difficult condition to satisfy experimentally.
2.3*2 Calculation of Stacking Fault Energy
Data obtained from x-ray line broadening and peak shift measure
ments can be used to calculate the stacking fault energy. In studies by
Smallman and Westmacctt (1957) and Otte and Welch (1964) the stacking
was obtained from the relationship
(13) where is a constant such that = 4k
independent of the properties of the material. G is the shear modulus
and aQ the lattice parameter. y(hkl) are mean squared strains
in the (hkl) direction determined from line broadening and Of is the
stacking fault probability from peak displacements. Equation (13) shows
that for a given alloy y should be directly proportional to
^ ( h k l ) ^ regardless of the type of deformation, since G, aQ , and
K(hkl) are constant. Adler and Otte (1966) show that l
^ (hkl)/^ can
34 ext o < t
Fig. 14. Typical cosO cotO versus cosO cotO curve for cold-worked filings.
35 vary with the mode of deformation. Thus they find that ^ (hkl)^^
for filings is different by a factor of about three than the same ratio
for drawn wires. From
(13) then it could be concluded that
varies with the type of deformation. Adler and Otte (
) regard X as
a more fundamental property, however, and one that helps determine the
dislocation configuration rather than one that is determined by it.
They therefore modify
) so that
is the same regardless of the
mode of deformation. They use the relationship
K ( h k l / ao ( f L X h k l / t f where is a proportionality constant between
^ and the width
the extended dislocation. The value of
depends upon the interaction
of the leading partials at slip plane intersections and the barriers
formed. With different modes of deformation there are different stress
distributions and different dislocation configurations. Consequently
different types of barriers can be expected to form at slip plane inter
sections. It is the factor ^ which takes into account the mode of
deformation. Adler and Otte (
) give a complete derivation of
J/ssang, Hirth, and Hartley (
) treat all of the extended dislocation barrier configurations that occur in the FCC structure.
These are, using Thompson's notation:
1. B(5 (d) + <5A(d) +
+cfB(a) = ($A +
Dc(+ d 6
2. B 6 ( d ) + 6A(d) + Do( (a) + d C(a) = B<5 + d C + tfD/Atf 12.54
3. B 6 ( d ) + d)A(d) + CCf (a) + Of = 6 a
+0fD + BC/a'6 17-65
+ d D/AG 3.22
B<5 (d) + ($A(d) + DQf(a) + (ZC(a) =
+ T + BD/^f
6. B 6 (d) + (^A(d) + B
(c) + ^D(c) = B 5 + B J' +
l 8 .44 (average)
The small letters in parentheses refer to the plane in which the partial
lies. Thus (d) refers to the plane opposite corner D of the tetrahedron.
Reactions 1-4 form symmetric barriers with barrier (l) being the Lomer-
Cottrell barrier mentioned previously. Reactions 3 and
Using anisotropic elasticity theory they have calculated the
values for each of these barriers for some alloys including some Cu-Zn
alloys of from 4.1 to 22.7 per cent Z n . The value of y- is in units of
dynes and is interpreted as the sum of the glide forces exerted by two
arms of an extended barrier on the third arm. Adler and Otte (
extrapolated the values of ^ for the copper-zinc alloys studied by
Jffssang et al. (
) to 30 per cent zinc. These extrapolated values
for the various barriers for Cu-30w/o Zn are given with the above list
For Cu-30w/o Zn the value
of j' =
l4erg/cm^ has been measured
by Howie and Swann (
) using the node method. Using this as their
anchoring value for
and the values for (hkl) ^ measured
using the x-ray method by Warren and Warekois (1935) for Cu-30w/o Zn
filings produced at room temperature, Adler and Otte (
Eq. (l4) to calculate a value for
for Cu-30w/o Zn filings. They
arrived at a value of 22.8 x 10 ^ dynes. Comparing this value with the
extrapolated values given for this alloy they conclude that filing
produces both symmetric barriers of Type 3 and asymmetric barriers of
Type 6. Using x-ray data for tensile loaded wire and the same
they find that tensile deformation produces barriers of Types 2 and 3
In discussing the application of
(l4) to x-ray diffraction
data, these authors point cut that when the stacking fault energy is
low, dislocation pile-ups tend to form and
(l4) can be modified to
take this into account. They modify
(l4) as follows: y 'Xhkp
_ n rz,2\
* ...... ?
/ ' ao y r F V (hkl/tf
where n is the number of extended dislocations per pile-up and F is a
factor which takes into account that there may be an interaction
between the strain fields of the dislocations. F is given by
F = [ln(Ro/ro ) + 0.776 n ] / i n d ^ A O . (l
Where R is the outer cutoff radius and r is the core radius. F is
o o generally in the range of 2 to 5* Adler and Otte (
) assume that the
dislocations piled up behind the barriers are dissociated screw disloca tions and use the room temperature value of ^ = 4.2 x
^ dynes which
they extrapolate from the data of J^ssang et a l . (
)• Using this
value of and the values of ^E'^Q^hkl1 d \ and
for deformed filings, Adler and Otte (
) obtain n/F = 6 using
)• Using F = 4, n is then in the range of 20 to 25 dislocations
In his review of the x-ray method of measuring the stacking fault energy, Otte (
) uses a value for
for Cu-30w/o Zn filings of
19 x 10 dynes as He takes is considered to be a
critical quantity in making the x-ray method more valuable. Only
estimates for are now available.
In calculating the quantity
and determining the types of
barriers formed, Otte (
) notes that the value of
sion electron microscopy is only thought to be accurate to within + 10%.
Since this was the anchoring value it affects the value of ^ as well.
The error in
from Eq. (l4) is thought to be anchoring value, probably about + 20%.
Although the transmission electron microscope is far superior to
the x-ray method for studying the details of dislocation structure, the
x-ray method is capable of measuring the effects of very high disloca
tion densities (such as in deformed filings) in much greater volumes of
material. The x-ray method takes into account the effect of large
numbers of stacking faults while the electron microscope can examine
only a few at a time. X-ray data are thus more meaningful in a
statistical sense and correlate well with mechanical measurements.
2.4 Temperature Dependence of Stacking Fault Energy
Many authors have shown that the stacking fault energy in alloy
systems is dependent upon composition. Vassamillet and Massalski (
found a decrease in stacking fault energy with increasing solute con centration in some binary inter-noble metal alloys between Au, Cu, and
Ag. In their x-ray studies they assumed that the stacking fault energy
is proportional to C( ^ and that this is valid for a constant disloca
tion density and configuration. The results of x-ray measurements by
Warren and Warekois (1955) on Cu-Zn alloys show this same decrease of
stacking fault energy with increasing solute concentration.
In a metal such as cobalt which has a HCP -> FCC transition the
stacking fault energy is clearly temperature dependent. This can be
seen with reference to F i g . 15 which is a plot of the free energies of
the two allotropic forms of cobalt as a. function of temperature. Since
the free energy of the hexagonal phase is lower below the transformation
temperature this phase predominates. The FCC phase predominates above
the transformation temperature. The stacking fault energy at any
temperature is equivalent to the difference between the free energies of
the two phases. The stacking fault energy is expected to be zero at the
Working with Co-Ni alloys, Ericsson (
) studied the tempera
ture dependence of stacking fault energy between 20° C and 710° C using
the electron microscope. His node measurements, made at the elevated
temperatures showed that the stacking fault energy decreased with
increasing temperature in the HCP phase and increased with temperature
in the FCC phase. Tisone (
) also reports an increase in stacking
fault energy with temperature in a Cu-l4w/o A1 alloy. His node measure
ments were made on samples quenched from various temperatures. The node
radius existing at the elevated temperature was assumed to be retained
by the quench.
Stacking Fault Energy
Free energy as a function of temperature for cobalt.
Smallman and Westmacott (1957) have made x-ray studies of low
temperature deformation in several copper and nickel alloys. They noted
about a threefold increase in the stacking fault probability in pure
copper when filing and x-ray analysis was done at liquid nitrogen
temperature rather than at room temperature. This effect was apparently
due to the non-equilibrium spacing of the partial dislocations and the
presence of pile-ups. When samples were allowed to warm to room tempera
ture the stacking faults annealed out (as measured by the stacking fault
probability) along with a major portion of the line broadening. As the
time at room temperature increased to
minutes faults continued to
anneal out and the stacking fault probability showed a steady decrease.
At the same time the measured strains (ill) showed a decrease as well, although the only values given are for the extremes of 0 and
minutes at room temperature . Their data do not show the change in
)/5f with temperature but as Eq. (l4) shows, a change in this
ratio would indicate a change in the stacking fault energy. The data of
these authors do show however that the final recovered line breadth of
the samples deformed at liquid nitrogen temperature is smaller than that
of a sample deformed at rocrr temperature, although greater than that of
a well-annealed sample. They note that since the dislocation configura
tion in copper filed at room temperature is observed to be piled-up, the
barriers that cause pile-ups of dislocations at liquid nitrogen tempera
ture must be dispersed when the sample warms to room temperature. Thus
in calculating the stacking fault energy in a sample filed at liquid
nitrogen temperature and allowed to warm to room temperature, a change
in the value of ^ in Eq. (l4) from those for deformed filings, might be
necessary. This would compensate for changes with temperature in the.
42 types of barriers and the number of piled-up dislocations.
Christian and Swann (
) state that in measuring the effect of
temperature on stacking fault energy the x-ray method gives only a
change in stacking fault probability which is related more to the
recovery kinetics on unequilibrium stacking fault widths than to any
with temperature. This would be true if
was the only
parameter measured. The measurement of both Of and (hkl ) ’ however, would seem to permit the calculation of
at various temperatures using
I l l . OBJECTIVES
The objectives of this investigation were the following:
1. The determination of Q( and (hkl) ^or Cu ~"30v//o Zn filed
at liquid nitrogen temperature, using the x-ray diffraction
2. The determination of these same parameters in the sample at room temperature and at 100° C .
3• The calculation of the stacking fault energy of the alloy at the different temperatures.
I V . EXPERIMENTAL PROCEDURE
4ol Specimen Preparation
■>.1.1 Melting and Casting -
The brass used in this investigation was prepared from 99.998%
Cu donated by the American Smelting and Refining Company and 99-999^ Z>n
provided by the. Consolidated Smelting and Refining Company of Canada.
The chemical analysis of these materials is given in Table 1. The
component metals were chemically cleaned and sealed in an evacuated
Vycor tube in the approximate ratio of Cu-JOw/o Zn. Melting was
accomplished by heating the tube in a muffle furnace at 1100° C for one
h o u r . The molten charge was thoroughly mixed by shaking the Vycor tube.
It was then allowed to cool in air. The tube was then replaced in the
furnace and the alloy was homogenized for 72 hours at 800° C . Total
ingot weight was approximately
4.1.2 Chemical Analysis of Cast Alloy
The brass alloy was analyzed using the x-ray fluorescence method.
the preparation of five standard solutions of known weight per cent zinc.
These solutions were prepared by first making separate solutions of Zn
and Cu in nitric acid to a concentration of 0.01 g/cc. A standard of
30w/o Zn was then prepared by adding 30 c c . of Zn solution to 70 c c . of
Cu solution. In this way standards of
, 29, 30, 31, and 32w/0 Zn were
prepared. The ratio of the Cu K-alpha intensity to the Zn K-alpha
44 ■ ■ .
Detailed Chemical Analysis of High Purity Copper and Zinc
Major Impurities a/o*
Ni As Te Sn
' 0.0002 0.0002
A1 ■ ;
99,999 + Zn
■*Reported by suppliers.
Parts per Million
Pb Mg .
0.1 1.0 0.1
intensity for these solutions was then determined using the x-ray
spectrometer„ This established a standard curve of the intensity ratio
Cu K-alpha/Zn K-alpha versus weight per cent Zn. The weight per cent Zn
in an unknown sample of the brass alloy, dissolved in nitric acid, could
then be found by determining the Cu K-alpha/Zn K-alpha intensity rela
tion and comparing this with the standard curve. This was done for
samples taken from the top, middle, and bottom of the ingot. The results were as follows: -
30-99 w/o Zn
30.68 w/o Zn
30,77 w/o Zn
30.81 w/o Zn .
The actual filings used in the stacking fault study were also analyzed
in this way and were found to be 30.79 w/o Zn. Diffraction conditions
for the fluorescence analysis are given below.
Tube Platinum Scintillation Counter Voltage 1400 .
Analyzing Crystal LiF
) has discussed quantitative analysis using x-ray
fluorescence in detail.
4ol°3 Preparation of Brass Filings
The cold worked Cu-30w/o Zn powder was produced by filing the
ingot while at.liquid nitrogen temperature. The ingot was clamped in a
holder inside a stainless steel container. The container was then
filled with liquid nitrogen until the ingot was submerged0 A Nicholson
mill-smooth file adapted for use in a sabre-saw was used for the filing
operation. Filing for approximately 20 minutes produced enough filings
for x-ray analysis.
The annealed standard required for the Stokes (1948) correction
for instrumental broadening was produced in much the same w a y „ Here,
however, the filings were produced at room temperature and annealed
under a hydrogen atmosphere for one hour at 4$0° C as suggested by
Warren (1959)• The annealed powder was stored in an evacuated dessicator
until used in the diffraction analysis.
4.2 X-Ray Studies
The General Electric XRD-5 x-ray unit was aligned prior to the
x-ray studies as outlined in the manufacturer's manual0 The absolute
zero of the instrument was set using a powdered sample of Q[ quartz.
The diffraction conditions used with both the cold-worked and annealed • samples were as follows:
Radiation Co K-alpha
Incident Beam Slit 3° MR
Diffracted Beam Slit .010 in.
Scanning Speed 0.12°/min.
Counter Tube Proportional
Counter Tube Voltage 1805 volts
Take-off Angle 3°
Diffrapted Beam Monochrometer— LiF crystal, 0.010 in. slit
Intensity values were recorded by a digital printer which printed each
0 . % ° of 29.
It is important in studies of this type to maximize the peak-to-
background ratio* This enables more accurate determination of the peak-
maximum .position's 'resulting in better experimental values of Of o It was
found that Co K-alpha radiation minimizes .fluorescence from the sample
and thus improves the peak-to-background ratio« Copper radiation was
found to be unsuitable due to the large amount of fluorescence produced
in the sample by the short wavelength portion of the continuous.
spectrum. A diffracted beam monochrometer using a doubly curved lithium
fluoride crystal was utilized as well. This also helped to.Increase the
peak to background ratio. A slow scanning speed, as recommended by
Warren (1959) was used to minimize errors. The take-off angle of 3° is
recommended by the manufacturer as providing good resolution as well as
4 o2 o1 Depressed T emperature Studies
The cold-worked filings were kept at liquid nitrogen temperature
throughout the diffraction analysis. The filings were left in the
filing container until only a small amount of liquid nitrogen remained.
They were then poured through a 35 mesh screen which was submerged in
liquid nitrogen. This screening served to remove particles of ice which
formed during filing. A small horseshoe magnet was used to remove
particles of steel from the brass powder. These particles formed as the
steel file deteriorated during filing. The filings were then.'transferred
to the sample holder for the diffraction analysis. The sample holder,
shown in Fig. l6, was used for both the liquid nitrogen temperature and
higher temperature studies. The tank which held the liquid nitrogen was
made of l/l6-in. brass sheet surrounded by 3/8-in. thick styrofoam for
insulating purposes. Tank capacity was approximately 400 c c . The front
face of the tank was 1/8 in. stainless steel provided with a 1-in. wide
opening for placement of the sample. A 0.020 in. thick section of
copper sheet was soldered into this opening in such a way as to leave a
depression l/l6 i n . deep into which the sample could be packed. The
copper sheet was thus between the powder sample and the liquid nitrogen
in the tank, providing a high rate of heat abstraction from the sample.
The sample holder was equipped with a micrometer adjustment which per
fractometer axis. This made it possible to keep the face of the sample
aligned with the.diffractometer axis in spite of the expansion of the
sample holder with temperature. Rotational adjustment of the sample
holder about the diffractometer axis was also provided.
sample holder was first immersed in a large tank, of liquid nitrogen with
the face of the sample holder in a horizontal position. This is shown
in F i g . 17. .Th e ■level of the liquid nitrogen was made just high enough
to cover the sample cavity. The mixture of liquid nitrogen and cold-
worked filings was then poured carefully into the sample cavity. A flat
steel blade, cooled to liquid nitrogen temperature, served to press the
filings into .the sample cavity. The surface of the powder sample was
then made completely flat and coincident with the stainless steel face
of the sample holder. An extremely flat sample surface is critically
Fig. 16. Sample holder for x-ray studies.
17. Sample holder in loading position.
important in assuring reliable results. When the sample holder was
raised to the upright position with liquid nitrogen in its tank, the
sample, remained packed in the sample cavity. The sample holder was then
mounted on the diffractometer and covered with a plastic envelope which
was sealed to the base of the sample holdero This arrangement is
pictured in F i g . l 8 . The purpose of the plastic covering was to prevent
the formation of frost on the sample by maintaining a dry nitrogen
atmosphere. The dry nitrogen was fed in through the base of the sample
holder and emerged through the small section of copper tubing visible in
Fig. l8. The dry nitrogen was cooled by passing it through coils cooled
by liquid nitrogen as shown in Fig. l 8 . The nitrogen stream was
directed at a point about 1 in. below the face of the sample.
In. order to prevent attenuation of the x-ray beam by the rela
tively thick (0.004 in.) plastic envelope, a window of very thin
(0.0005 in.) plastic was used. Both the incident and diffracted beams
passed through such windows. Any decrease in intensity was not
measurable. The outside of the plastic envelope and window was kept
free of frost and moisture with streams of warm air from electric
blowers. Liquid nitrogen from a pressurized flask was used to replenish
the supply in the sample holder tank. This entered the tank at the top
through a copper tube. Once the sample holder was in position on the
diffractometer, approximately a half an hour was allowed for the tempera
ture to stabilize. This minimized any chance of expansion or contraction
occurring during the diffraction scan. This is an important point since
any shift of the sample holder face causes a shift in.the 20 position of
Fig. 18. Sample holder in position on diffractometer.
the diffraction peaks. This' would in turn affect the calculated value of
4.2.2 Elevated Temperature- Studies
and (200) peaks had been obtained the liquid nitrogen in the tank was
allowed to boil away. After a period of about two hours the sample was
considered to have reached room temperature and the dry nitrogen flow
was stopped and the plastic envelope removed. The diffraction scan at
room temperature was then carried out. The sample was not removed or
disturbed in any way. Except for the change in temperature the experi mental conditions were unchanged.
The analysis at 100° C followed.. This temperature was achieved
by the simple expedient of boiling water in the sample holder tank. An
immersion type electric heating coil was used with water added to the
tank at the rate of about 10 ml/min. The water was allowed to boil for
a half-hour before the diffraction scan was begun. This allowed the
entire sample holder to adjust to the new temperature thus minimizing
Later a second sample was filed at liquid nitrogen temperature
and then allowed to warm to room temperature for two hours. A diffrac
tion scan of the (ill), (200),.(222), and (400) peaks was made. A
diffraction scan of this sample was also made at 100° C . This second
sample was used as a check on the data, for room temperature and 100° C
obtained from the analysis of the first sample. A screen analysis was
carried out on both samples when the diffraction scans were complete.
In order to use the Stokes (1948) method of removing the
instrumental broadening it was necessary to complete a diffraction scan
of the annealed standard at each of the three different temperatures.
Identical diffraction conditions were used for the cold-worked and
annealed samples. The same sample loading procedure' was not necessary
for the annealed standard. The only requirement is that both the
annealed and cold-worked samples be studied at ,the same temperature and
under identical experimental conditions.
4.3 Analysis of Data
The Fourier cosine coefficients
were obtained using the
Stokes (1948) relationship. The actual computation was accomplished
using a program compiled for the GDC 6400 computer. This program was
written by Aqua (1964). The program removes the background intensity
and the K-alpha-2 component of the K-alpha doublet by the Rachinger
(1948) method. The positions of the peak maximums are located by a
three-point parabola fitting routine. Simmons (
) presents this
program in detail. The Fourier coefficients were recorded on standard
IBM cards for use in a second computer program.
This second program, written by Cotter (
) calculates the
mean squared strains %y) (hkl)' an(^ the stacking fault probability
from the Fourier coefficients and the peak positions, respectively. The program plots Eq. (
) analytically giving values for the intercepts at
2 . PF 2 hQ = 0 which are A£ . The slopes of the curves for In A^ versus h Q are also computed and correspond to (^kl)* ^he effective particle
sizes can be determined by extrapolating the linear portions of the
PF versus L curves to their intercepts on the L axis as shown in F i g . 13•
The stacking fault probability Qf was calculated using Eq. (10).
Initially the experimental peak positions were converted to a
values which were plotted versus cos© cot©. Then a straight line was
fitted to the points a' and a” at f '(©) and f"(©), respectively. This
gave values of which were then used in Eq. (ll) for the determination of A a (hkl)*
The stacking fault energy calculation was not included in the
computer program. The stacking fault energy was calculated for the
three different temperatures using Eq. (l4) . The value of used was * The (^ q q
) value does not give reliable
results because of the extremely broad (400) peak normally found in
cold-worked materials as pointed out by Otte (
)• The value for
L = 50 A was used because this is the value commonly reported and used
in the literature. The value of Of used in Eq. (l4) was the average
(ill) + ^(200)] //2‘ The values of Of obtained from the (222)
and (400) peak positions are less accurate as pointed out by Warren and
Warekois (1955), since these peaks are extremely broad and there is
considerable uncertainty in the location of the peak maximum position.
The values of the constants
, K, a^, and F which were used in the
actual calculations are reported in a later section.
V. RESULTS AND DISCUSSION
The values of the stacking fault probability,
, were calcu lated using Eq. (10) and the experimentally determined peak positions.
The rrr.s strains were calculated from the slopes of the In AT vs. h
L o lines at L = 50 A using Eq. ( 9
)• The stacking fault energy was obtained
from Eq. (l4) for each of the three temperatures. The number n of dis locations in a pile-up was obtained using Eq. (
). These results, as
well as values for
and a^ used in calculating the stacking fault
energy, are presented in Table 2. The room temperature results of
Warren and Warekois (1955) and Wagner and Helion (
(qpp) are shown also for purposes of comparison. The value shown
as Cf is equal to + ^ ( 2 0 0 ) ] ^ ’ The lattice parameter aQ is the value determined from the individual values for the four reflec tions studied.
5.1 Low Temperature Results
The stacking fault energy of the ccld-worked filings at 100° K
2 was found to be 16.8 erg/cm . This is in good agreement with the room temperature electron microscopy results of Howie and Swann (
16.2 erg/cm . In calculating the stacking fault energy the value of
K ( n i ) =
was used as recommended by Otte (
) . Since
directly with the shear modulus G, the value of ^* = 19 x 10 ^ dynes
recommended by Otte (
) for room temperature calculations, was
adjusted for the change in G with temperature. The measurements of the
Results of Diffraction Analysis
300° K #1 a(in)
^ (111) x 10
300° K #2 0.0144
373° K #1
373° K #2
^(dynes x 10^)
/ (erg/cm2) n
Warren and Warekois
(1955) a = 0.023
= 12.96 x 10"6
Wagner and Helion
low temperature elastic constants of some alpha brasses made by Rayne
(l959)i and extrapolated to 30w/o Zn were used. The value of
x 10 ^ dynes was then multiplied by G (^oo) °K//G(300) °K
= 2 0 03 x 10 ^ dynes for the low temperature calculations. Both
/^ (l l l ) ^ and the individual Of and ^^.i) values agreed
closely with the room temperature x-ray results of Warren and Warekois
(1955) and Wagner and Helion (
)• It seems somewhat surprising that
the individual values of Of and are not considerably higher
than those reported by these authors since the deformation here was at
liquid nitrogen temperature. As noted previously, Smallman and
Westmacott (1957) found that
and (hkl) increased with low
temperature deformation. This effect can be explained, however, on the
basis of the particle size of the filings and is discussed in a later
The value of n ~ 18 dislocations per pile-up was calculated
erg/cm^ and 4.4 x 10 ^ dynes for a dissociated screw dislocation. F was taken to be 4 as suggested by Otte (
). He points out that F is usually in the range of 2 to 5*
5• 2 Room Temperature Results
The average value of the stacking fault energy from the two room
temperature determinations was
erg/cm2 , only slightly higher
than the low temperature value. This indicates little or no change in
ture values are also in close agreement with the room temperature electron microscopy results of Howie and Swann (
). The values of
Of and (2.1]) ^or the two room temperature measurements were in
close agreement with each other indicating good reproducibility at this
temperature. The measured values of both Qf and ( m ) showed a
decrease when the sample was warmed to room temperature in agreement
with the results of Smailman and Westmacott (1957)• The ratio
/> (]_] I)/6-*' showed a slight increase in both of the room temperature
samples. In the calculation of
the value of ^ was adjusted to
compensate for the change in the shear modulus in going from 100° K to
room temperature. The value of n determined from Eq. (15) using
= 4.2 x 10 ^ dynes for dissociated screw dislocations did not change
in. going from 100° K to 300° K.
5*3 Elevated Temperature Resul ts
No value for (q
^) for the first elevated temperature sample is given since the measured value was less than the minimum value of 10 allowed by the computer program. A considerable decrease from the room temperature value would be expected, however, due to the stress relieving effect of the elevated temperature. The value for the ^100J
direction was 40 x 10 and is the one given in Table 2.
The measured value for
for this sample showed an increase in
the faulting probability over both the 100° K and 300° K measurements.
This is a questionable result since other studies (Wagner, 1957;
Smallman. and Westmacott, 1957) have shown that faults produced at liquid
nitrogen temperature anneal out at or slightly above room temperature.
A further decrease in O' from the room temperature value is to be
The second sample was analyzed at 100° C sjid again the results
appear anomolous. While the faulting probability in this sample does
show a decrease from the room temperature value, the measured rms
strains were found to be slightly higher than those at the other
temperatures rather than lower. For the first sample, (jf seems unreasonably high, while for the second
unreasonably high. This situation makes the validity of any calculated
value for the stacking fault energy at this temperature very questionable.
Calculations based upon
40 x 10 ^ for the ^10oJ directio]
The measured value of
/ = 0.29
for sample No. 2 yields since the rather than first sample is based upon not strictly comparable to the other measurements. The two values at
this temperature differ greatly, however, and it is felt that neither
can be considered completely reliable. Since n, the number of disloca tions in a pile-up is calculated using these values for y^, its relia bility is affected as well.
5 Discussion of Results
As mentioned previously, Smallman and Westmacott (1957) found
that low temperature deformation of pure copper produced a much higher faulting probability
and room temperature deformation. Filings formed at room temperature had a faulting
62 probability of
= 3»3 x 10 ^ while in those formed at liquid nitrogen
increased to approximately 12 x 10
The fact that the stacking fault probability for the sample
filed at liquid nitrogen temperature in this study was not considerably
higher than that measured by Wagner and Helion (
) who filed at room
temperature, may be explained on the basis of the particle size of the
sample. Anantharaman (
) found that as the size of filed particles
increases the stacking fault probability and the x-ray line breadth
decrease. He concludes that the larger particles have undergone less
deformation and thus have lower stacking fault densities than the
smaller particles. For Cu-33w/o Zn filed at room temperature he found
= 0.039 when the filings were all less than
mm. For a
sample with particles all greater than 0.22 mm the stacking fault
probability decreased to 0.029• At the same time the line breadth also
decreased. Since residual lattice strains contribute to the breadth of
x-ray reflections it is expected that (hkl) decreased as well.
A screen analysis of the samples used in this study showed that
while only 2.4w/o of the sample had particles greater than
1 1 .4w/o was greater than 0.147 mm and 57»lw/o was between 0.033 mm and
0.147 mm; 31«5w/o was less than 0.053 mm. Since the sample was loaded
while the sample holder was in the horizontal position it would be
expected that the smaller particles would settle toward the back of the
sample cavity leaving the larger particles nearer to the surface. Thus
the small particles would contribute less to the total diffracted beam
than the larger particles nearer to the sample surface. The data of
Wagner and Helion (
) were based upon a sample with particles less
mm. For these reasons the stacking fault probability and rms
strains for the sample at 100° K are not considered to be inordinately
The value for ^ at 300° K was adjusted for the change in the
shear modulus in going to room temperature but not for any change in the
nature of the barriers present. Although the collapse of some of the
weaker barriers with increasing temperature is expected as pointed out
by Small man and Westmacott (1957
and Otte (
), the relative
strengths of the various barriers are not well known. Also since the ^
value used corresponds to barriers of Type 3 and Type 6, and is
virtually the largest that can occur, it does not seen that the collapse
of some of the barriers would significantly affect the value of
Barriers with higher ^ values have been postulated (Teutonico, 1964)
but have not been treated in detail. Thus it seems unwarranted to make
any assumptions regarding a change in with possible barrier collapse.
The decrease in Of and \ ^ n ) which occurred with warming
to room temperature is in agreement with the results of Smallman and
Westmacott (1957) and Wagner (1957)• The decrease in with
increasing temperature is associated with some barrier collapse at the
head of piled-up groups as well as the rearrangement of non-equilibrium
dislocation configurations such as tangles into arrangements with lower
values of residual stress. The partial recombination of dissociated
dislocations is responsible for the decrease in C( with increasing
temperature. A decrease in the width of stacking faults results in
less total faulted area and a lower faulting probability. The important
point is that the decrease in the faulting probability obtained in this
study seems to be due to thermally activated processes rather than to
any increase in the stacking fault energy with temperature. The
stacking fault energy values obtained for 100° K and 300° K differ only
slightly and this difference is not considered significant. The driving
force for the decrease in the widths of faults is supplied by the
internal stress. As the temperature is increased the internal stress is
thought to be sufficient to overcome the resistance to the motion of
dislocations provided by jogs and forrest dislocations. The decrease in
the faulting probability thus seems to be associated with the amount of
deformation that can be introduced and retained at depressed tempera
tures. The increased temperature allows the partials to assume a
It is unfortunate that the ancmolous results for Qf and
the first and second elevated temperature samples do not
permit the calculation of reliable values for the stacking fault energy.
The increase in G for the first sample at 100° C may have been caused
by a slight shifting of the face of the sample during the analysis.
Although great care was taken to insure that this did not occur, it
seems to be the only reasonable explanation.
An indication of the decrease in the value of (-^ i
300° K to 373° K is found in F i g . 19. Here an average line has been
drawn through the actual diffractometer trace for the (ill) reflection
from the sample at 300° K and 373° K . The two (ill) profiles were then
superimposed. The breadth of the peak for 100° C is not as great as
that for the sample at room temperature. This decrease in breadth is
due mainly to the large decrease in the lattice strains,
Figure 20 compares the peak breadths for the same reflection, of the
sample at 100° K and room temperature. While the room temperature peak
is slightly narrower, the data show that the rms strains did not
decrease nearly as much going from 100° K to room temperature as from
room temperature to 373° K . Since small domain size and faulting
effects also contribute to line broadening, as discussed previously, it
The extremely high value for for the second sample
at 100° C seems to be due to some experimental error. It is not known
exactly what caused the rms strains to be so large but any change in the
contour of the sample face could have contributed to this error. The
large decrease in (hkl) no^ec^ the first sample at 100° C is thought to reflect the true situation. An increase in ^ \ with temperature must be Time limitations precluded a further investigation of the 100° C temperature .
Fig. 20. Comparison of (ill) profiles for 100° K and 300° K .
V I .. CONCLUSIONS
The results of this study permit, the following conclusions to be drawn.
1. The stacking fault energy of Cu~30w/o Zn deformed at liquid'
nitrogen temperature does not appear to change significantly
when the temperature is raised from 100° K to
° K . The
decreased stacking fault probability and recovery effects noted
with the increased temperature are considered to be caused by
the unpiling of some piled-up groups, the collapse of some weak
dislocation barriers, and an increase in the size of the
coherently diffracting domains» The narrowing of stacking fault
ribbons as a result of the thermally activated processes
described, and not an increase in the stacking fault energy,
also contributes to the recovery effects.
2o The recovery effects noted when the temperature was raised to
room temperature are associated mainly with decreased, faulting
and. an increase in the size of the coherently diffracting
domainso Raising' the temperature from 300° K to 373° K brought
about the decrease in the rms.strains previously noted»
) analysis of the x-ray method for determining the
stacking fault energy has been used as a basis for calculations of the
stacking fault energy in this study. Prior to the publications by Adler
69 and Otte (
) and Otte (
) the mode of deformation and the barriers formed were not considered in stacking fault studies.
The present study was conducted using Cu-30w/o Zn in order to •
confirm the reliability of the method as there is considerable published
data for this alloy. The low temperature sample holder developed for
this study permits an extremely flat sample face to be formed while the
sample is at liquid nitrogen temperature. Another advantage is that no
window is required to cover the sample face. This results in a more.
intense incident and diffracted beam. In its present state the sample
holder can be used to attain temperatures up to about 170° C utilizing
boiling liquids. Soldered connections in the tank portion will fail
above this temperature. This sample, holder should be very useful, in
further stacking fault studies.
The x-ray method of measuring the stacking fault energy seems to
be the best technique for measuring this parameter at depressed tempera
tures. It would be difficult indeed to introduce low temperature cold
transfer and maintain the sample at the depressed temperature. Further
more there is no low temperature tilting stage available for the electron
microscope thus making the examination of such a specimen very difficult.
The x-ray method also permits the measurement of the stacking fault
probability, a parameter unique to this method. This is a meaningful
parameter and gives important information as to. the.density of stacking
faults in a particular sample.
VII. SUGGESTIONS FOR FURTHER WORK
It should be useful to study the reproducibility of data
obtained from seme common FOG metals and alloys filed at room tempera ture. Prior to the present study values of
obtained for 325 mesh Cu-30w/o Zn filings formed at room temperature.
These values correspond very closely to those of Wagner and Helion
) for the same alloy. Room temperature filing and x-ray examina
tion eliminates the tedious and delicate experimental work required for
studies at liquid nitrogen temperature.
A further investigation into the effects of elevated tempera
tures upon cold work introduced at liquid nitrogen temperature should be
carried out. Specifically the 100° C temperature should be studied in
an effort to determine the source of the errors encountered in this
study for that temperature.
A study of the effect of particle size on the stacking fault
probability and rms strains should be made. While Anantharaman (
has investigated this, further studies should be made. His measurements
were of C( and the line breadth and (hkl) was thus only measured
indirectly. Since small domains also contribute to line broadening it
should be interesting to study the effect of the particle size upon both
residual strains and the size of the coherently diffracting domains.
It was found to be practical to screen a sample under liquid
nitrogen. Although only a 35 mesh screen was used in this study, finer
screens could also be used. Thus the particle size could be made at
liquid nitrogen temperature. This has never been done before.
In order to utilize Eq. (l4) for the calculation of the
stacking fault energy from x-ray data it is necessary to have values
for the force factor ^ . The values of J^ssang et a l . (
) were used
in this study. For other FCC metals and alloys it would be necessary
to calculate values of ^ for the various barriers using anisotropic
elasticity theory. This could be done for an alloy that has not been
studied before. Subsequent x-ray analysis would yield measured values which could be used to calculate the stacking
fault energy. The electron microscope could be used as a check on such
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