IN A Cu-30w/o Zn ALLOY by

Craig M. Shevlin

A Thesis Submitted to the Faculty of the


In Partial Fulfillment of the Requirements

For the-Degree of




1 9 6 9


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.



This thesis has been approved on. the'date shown below:


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



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 • « , .......... ..


: 6l

68 VI, CONCLUSIONS . . . . . .................. , , , , , , , ,

V I I . SUGGESTIONS FOR FURTHER WORK . , .....................

SELECTED BIBLIOGRAPHY . , ....................... . . . . . .



Page vi viii ix



5 l8




3 8












57 r ' .

v / 4 . \ /.



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

FCC structure

« 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)

^ ?

......... . . . . . . . . . . . . .

13- Typical

PF versus L curve for determining effective



13 l4 l6

18 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









. 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 „ . . „





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.



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.


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 [110] 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

fault energy.

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

the partials.

Fig. 1. Dissociation of a unit edge dislocation into partials in the

FCC structure.




> b unit



> a


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

FCC structure.



b uni

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

the partials.

F i g . 3. Dissociation of a dislocation of mixed orientation into partials

in the FCC structure.

Repulsion fore

of partials


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


tetrahedron. A


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

FCC structure



--L--- -






^ V








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

probability of


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

these methods.

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

Delavignette (



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.

Warren (


) 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 (


) state

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.


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


(hkl) (

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 (


) bears

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




Cotter (

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:

(hkl) =



-d(ln A^)

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

on the

line is given by: a (hkl)1 =


a + m cos ©

sin© (12)


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

fault energy


was obtained from the relationship


(13) where is a constant such that = 4k

^ ^



and is

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

Eq. (14).

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:

Reaction Barrier

1. B(5 (d) + <5A(d) +

DCf(a )

+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






4. b




+ d D/AG 3.22


B<5 (d) + ($A(d) + DQf(a) + (ZC(a) =



+ T + BD/^f

11.03 (average)

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


form asymmetric


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

of barriers.

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 (


) used

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


given previously

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

per pile-up.

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


from transmis­

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

transformation temperature.

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


/ Temperature


Fig. 15.

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

change of


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

Eq. (14).


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.



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 ■ ■ .


Table 1

Detailed Chemical Analysis of High Purity Copper and Zinc



99-999 +.Cu




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.





Receiving Slit

Counting Time

0.01 in.

1000 sec.

Tube Platinum Scintillation Counter Voltage 1400 .

Analyzing Crystal LiF

Valenzuela (


) 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

KVP 50

MA 12

Incident Beam Slit 3° MR

Diffracted Beam Slit .010 in.


Scanning Speed 0.12°/min.

Counter Tube Proportional

Counter Tube Voltage 1805 volts

Gain 9-

Take-off Angle 3°


.. E



E out

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

high intensity.

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

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

positioning errors.

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.


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 (


) for



(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 (


) of


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


Table 2

Results of Diffraction Analysis


100° K

300° K #1 a(in)











^ (111) x 10





300° K #2 0.0144




373° K #1




0.4 |

(^50%100) 3.685

373° K #2






^(dynes x 10^)






/ (erg/cm2) n









Warren and Warekois

(1955) a = 0.023


) (ill)

= 12.96 x 10"6

Wagner and Helion



) a











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

narrower spacing.

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

) from

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 .

100° K

Fig. 20. Comparison of (ill) profiles for 100° K and 300° K .



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.


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


and were

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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