Carlos G. Valenzuela

A Thesis Submitted to the Faculty of the


In Partial Fulfillment of the Requirements

For the Degree of



In the Graduate College




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 acknow­ ledgment 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:

O' t4r J . Berner

Professor of

Metallurgical Engineering


The author is especially indebted to his advisor.

Dr. L. J. Demer, for his interest and helpful suggestions

throughout the investigation.

Special thanks are also due to A. W. Stephens who

helped greatly in the texture study.

The high purity copper donated by American Smelting

and Refining Company is gratefully acknowledged.

The assistance of my sister, Sylvia, in plotting of

pole figures and typing of the rough draft of the manuscript

is also appreciated.





LIST ^5K^5I?CPLE^5#oaoo#***@*oo*#**[email protected]@*#8oo*o#[email protected]@ooooVila

LIST OF TABLES, .................................. ix


INTRODUCTION.................................. 1 I

II THEORETICAL BACKGROUND,....................... 5

2.1 Rolling Textures in FCC Metals.......... 5

2.2 • 12

2.3 Dependence of Texture on Stacking

Fault E n e r g y 23

2.4 Effect of Solute Additions on

Interfacial Energy.................... 25



OBJECTIVES.........___ ..............____ 28

MATERIALS AND PROCEDURE....................... 29



RESULTS...............____ .................... 33

5.1 Twin Boundary Energies.................. 33

5.2 Rolling Textures........................ 34

5.3 Dependence of Texture on Stacking

Fault E n e r g y 38

D I S C U S S I O N 38

6.1 Twin Boundary Energies. .......... 38

6.2 Effect of Solute Addition on Twin

Boundary Energy....................... 48

6.3 Twin Boundary Energy vs Stacking Fault

E n e r g y 51

6.4 Texture Intensity Ratios................ 58

6.5 Effect of Stacking Fault Energy on

Rolling Texture. ...... 59 iv

Page e o o o e e e o e e o d e o e e o o o o e o o o o o e e e o o o o o a o o o o 6 4

e o o o o o o e o o e o o w o o o o o o o o o o o o o o o o o o o o e 6 6 *VIE3E™E

Z ^ P i ! P E j f 0 2 Z ^ [ 0 E l S o o o o o o o o o o o o o o o o o o o e e o o o o o o e e e o o y o o o o o o o o o o o 6

















Ball model of (a) splitting of total

dislocation into partials; (b), (c)

au ^

Distribution of values of twin/grain boundary

energy for pure copper and 1.03 w/o zinc in

drass....................................... 7 8

Distribution of values of twin/grain boundary

energy for 2.60 w/o zinc and 5.30 w/o zinc

rifi brass..................................... 79

Distribution of values of twin/grain boundary

energy for

6 .8 5 w/o zinc and 12.88 w/o zinc

in bras 8,0

Variation of mean and range of confidence with

number of values of twin/grain boundary

energy measured............................. 8l

Twin and grain boundary energies for Cu-Zn alloys......................................


(ill} pole figures of high purity copper and

and brass rolled

9 8 .3^. Weight percent

zinc in brass as indicated, a {.110? <112^

A [53li <001>...................... 83

{llli pole figures of high purity brass


9 8 .3/. Weight percent zinc in


brass as indicated. n{1103 <112> A {5315


1 # ^

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


{ill! pole figures of brasses rolled

96 ^.

Weight percent zinc in brass as indicated.

(After Merlini and Beck, 1955)..............


Variation of texture intensity ratio with weight percent zinc in brass.-............... 86 vl


Page Figure

11 Partial {llll pole figure scan of 99„


copper rolled l6 fo a.


8 f

12 Partial fllll pole figure scan of 99.999$ copper rolled

9 8 .3$.........................


13 Apparent stacking fault energies for copperzinc alloys.................................


14 Variation of texture intensity ratio with stacking fault energy....................... 90


Sketch Page

1 Surface tension and partial dislocation repulsion vs distance of separation of partialS

» e 6 » e e e e o e » o o o o e o o o o 6 e o e e o o e e e o » o o e o



Burger's vector relations on {llll plane of

FCC crystal a [email protected]@&@aaaoaooooaoo«»oooaaoaaaaaaaa 16

3 Dislocation node containing faulted areas.*,,.


^4 Twinned C r y s t a l 21

5 Mechanical analogy of twin surface energy...*. 21


Stacking fault with constriction.............. 23


Specimen mount for electropolishing........... 32


Formation of an annealing twin................ 42


Some observations of twinned grains........... 43

10 Schematic plot of Fermi energy vs {lll^ d-spacing across a stacking fault........... 57 viii







Detailed chemical analysis of high purity coppex* and z*Lnc 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Twin Boundary energy data,0««.».,..». = o.«.«»,e 72

Texture parameters and stacking fault



The method of obtaining twin boundary energies by

measurement of dihedral angles at twin-grain junctions can

be extended to alloys. Twin boundary energies can be used

as a guide to stacking fault energies. This method is

more reliable for estimating stacking fault energy in metals

with higher stacking fault energy than the method involving

measurements of radii of dislocation nodes due to the limit

of resolution of the electron microscope.

Stacking fault energy in low zinc brasses does not

decrease in a simple exponential manner with increasing

zinc as is usually indicated by extrapolated values from

brasses of higher zinc content. This energy actually

decreases rapidly for the first additions of zinc up to


1 weight percent after which the rate of decrease is

lower. ,

The copper-to-brass texture transition is shown to

depend on a decrease in stacking fault energy.



Studies related to the nature and determination of

stacking fault energies have gained prominence in the last

few years due to the important role stacking faults are

believed to play in the plastic deformation characteristics

of metals, A stacking fault can be thought of as a region

of HCP structure in a FCC structure, or vice versa. In a

close-packed structure, there are three sets of positions:

A, B, and C for the atoms in each layer. The stacking

sequence ABCABO... is characteristic of FCC structures

while the sequence ABABAB... is characteristic of HCP

structures. In a FCC metal, slip can occur on a {lll\

plane with the possibility of producing a stacking fault

in such a way that a B layer following an A layer of the

regular sequence becomes a C layer:





A usual stacking







A stacking fault (intrinsic)

Another of the types of stacking faults which can occur

is the annealing-twin fault which can be formed during

grain growth or recrystallization:










Annealing-twin fault

Stacking fault energy may be measured by essen­ tially three methods: (a) electron microscope observation

of dislocation nodes, (b) x-ray measurements of peak shifts,

and (c) measurement of coherent-annealing-twin dihedral


Measurement of dislocation nodes is limited to

alloys with a minimum of about

10 percent zinc in brasses

due to limits of resolution of the electron microscope


1 9 6 3 )0 The x-ray method yields fairly good

results for the frequency of stacking faults but it is

difficult to obtain stacking fault energies from this

frequency because values of dislocation density are re­ quired, At present, dislocation densities can only be estimated in metals because of the high density, about

IT p

10 dislocations/cm for cold-worked metal. The method involving measurement of annealing-twin dihedral, angles seems promising because it yields the ratio of twin boundary energy to grain boundary energy. The stacking fault energy is assumed to be twice the twin boundary energy (Read, 1953), Although this method has been used only for pure metals (Pullman, 1950j Inman and Khan, I 9 6 I;

Bolling and Winegard, 1958), it may be possible to extend

this method to alloys.

Addition of a solute to a pure FCC metal lowers

its stacking fault energy (Howie and Swann,

1 9 6 1 , for

instance). Lowering the stacking fault energy has been

suggested as a means of changing the deformation mode

through the change in cross-slip phenomena (Seeger,

1 9 5 9 J

Thornton, Mitchell, and Hirsch,

1 9 6 2 ). Since a change in

cross slip would change lattice rotations during plastic

deformation, it would be expected that this change could

be shown by a change of preferred orientation, or texture,

of the material.

Several papers have been written on the existence

and types of preferred orientations, or textures, in

metals and alloys (Hu and Goodman,

1 9 6 3 j Merlin! and Beck,

1955; Hu, Sperry, and Beck, 1952; Dillamore and Roberts,

1964). Because of the pronounced anisotropy produced by

some textures, a practical importance exists in under­ standing the fundamental causes of textures so that even­

tually desired formation or suppression of these can be


Mechanisms operative in the formation or altera­

tion of textures are somewhat elusive because of the

difficulty of separating the experimental variables that

contribute to the formation of these textures. However,

it is known that textures are the result of processing

the original ingots, alloying additions, plus methods and

temperature of deformations. In the processing of the ori­

ginal ingots, the type and degree of texture is a function

of the casting and subsequent heat-treating process.

Alloying additions seem to cause a change in texture from

the pure metal which is the same regardless of the alloying

element, provided that the second element is added in

solution in sufficient concentration (Smaliman,

1 9 5 5 ).

The temperature of deformation seems to affect texture

changes in about the same way as alloying additions (Hu

and Goodman, 1 9 6 3 ).

The type and degree of texture can be determined

by means of pole-figure constructions. The data for these

constructions are obtained by x-ray scans of the material

at several specimen-to-beam orientations.

Since cross slip is a common factor to both the

stacking fault energy and texture in a metal, it would

be expected that a change of stacking fault energy by

alloying would be indicated by the change in rolling tex­ ture observed.

This suggestion has been made by Dillamore,

Smaliman, and Roberts (1964).




„1 Rolling Textures in PCD Metals

When a metal is deformed by rolling, the grains

tend to rotate such that the resolved shear stress acting

on them is reduced. This causes a majority of grains to

approach the same orientation.

The material then possesses

a preferred orientation, or, a crystallographic texture.

A texture is usually detectable in a metal after about 20


30$ deformation but contains some texture due to casting

and processing even before deformation.

It is usually

necessary to deform to amounts greater than

90$ in order

to study conveniently the deformation texture since below

this approximate reduction some of the previous texture is

still present. The texture may be described quantitatively

by use of pole figures.

These figures, which are stereo-

graphic projections of the poles, or normals, of the crys­

tallographic planes, represent the orientation distribution

of crystallographic planes in the grains of the metal.

By analyzing the character and details of the pole figure,

it is possible to describe the degree and type of texture

that is present.

A pinhole x-ray photograph is an x-ray diffraction

pattern of a polycrystalline specimen obtained with a Laue


camera using monochromatic radiation. When such a pattern

is obtained, the Debye, or diffraction, rings which appear

are the result of an aggregation of spots formed by diffrac­ tion of certain crystallographic planes. Each spot corres­

ponds to a grain which has its planes so oriented to satisfy

the given Bragg condition.

In a metal whose grains are

oriented randomly, the population of spots around the cir­ cumference of a Debye ring is equally distributed.


the metal contains a texture, the Debye rings are of non-

uniform intensity or even discontinuous around the circum­

ference. The low intensity or missing arcs are due to few

or no grains being oriented at those positions for Bragg

reflection. The degree and type of texture may be deter­

mined by this method if pinhole photographs are taken at

several angles between the incident beam and the specimen.

A diffractometer counter tube measures the intensity

of the diffracted x-rays from a portion of the Debye ring.

In order to use this instrument for determination of

preferred orientation of a rolled sheet specimen, the

specimen must be rotated perpendicular to its rolling face

(beta angle) so that the entire Debye ring (or arcs in the

case of a textured specimen) passes through the counter


In addition to this rotation, it is necessary to

rotate the specimen about its rolling axis (alpha angle).

At zero degrees alpha the plane of the specimen bisects

the incident and transmitted diffracted x-ray beam.

Rotation of beta through 360 degrees at this alpha angle

yields the texture distribution at the periphery of the

pole figure „

5 degrees

and re-rotation of beta then yields the texture distribution

at 5 degrees in from the periphery of the pole figure. By

obtaining x-ray scans such as these for the alpha range from

zero to

9 0 °, the orientation distribution of crystallo-

graphic planes for the entire specimen can be found. The

diffractometer goniometer is set at a constant Bragg angle,

usually the angle to diffract (

111} planes in FCC metals.

Stereographic analysis can then be applied to obtain the

crystallographic plane most predominant at the surface of

the metal and its direction which coincides to the rolling

direction. This plane and direction are usually reported

as { hkl] <uvw>.

Since it has been well established that slip occurs

on close-packed planes, as the {ill]

in FCC metals, it

should be possible to predict the type of texture that will

occur. However, this has proved to be difficult because

of the interaction of slip systems and the impedance to slip

at grain boundaries and at dislocation entanglements.

An attempt to predict the rolling texture of FCC

structures by Boas and Schmid (1931) assumed operation of

three slip systems and derived stable end positions. They

correctly predicted a {lio} <

112 > for both tension and

compression textures for alloys but did not explain how the

crystals which were not initially in the end positions

could reach these positions. Taylor (1938) used a mathe­

matically rigorous analysis which predicted the simultaneous

operation of three to five slip systems. However, besides

the limitation of his assumption of homogeneous deformation,

the computations for a metal are very difficult.

Pickus and

Mathewson (1939) assumed unequal participation of the three

most favorable- glide systems. The most favorable systems

were defined as those in which the product of the resolved

shear stress and the cosine of the angle between the glide

direction and the direction in which free flow may occur

are the highest.

The above three theories all depend on the resolved

shear stress on the most favorable system to become greater

than the critical value until the resolved shear stress on

less favorable systems becomes high enough for slip to occur.

Boas and Hargreaves (1948) suggested that multiple

slip systems are required to operate only in the immediate

vicinity of the grain boundary, while in the body of the

grains slip occurs only on one or two planes.

On this

basis, they concluded that the deformation of the individual

grains in a polycrystal could be considered as being closely

similar to the deformation of single crystals.

Since this

theory simplified the analysis greatly while still predict­ ing the textures correctly, it was perhaps a major contri­ bution to the theory of rolling textures.

Two major difficulties still remained however.

These were (a) the resolved shear stress on the primary,

or most favorable slip system, still had to exceed the

critical value before the resolved shear stress on a less

favorable system became high enough for slip to occur and

(b) the different type of texture observed in pure PCC

metals could not be explained by any of the theories pro­ posed at that time.

The first successful attempt to explain the former

difficulty was made by Calnan and Clews (1950) who intro­

duced an "effective stress axis" which was removed from the

true stress axis by constraints imposed by surrounding


Since the effective stress axis did not coincide

with the true axis, the resolved shear stress on the most

favorable system would now be lower and thereby obviate

the necessity of exceeding the critical value.

One of the most recent and better analyses of

rolling textures is that of Dillamore and Roberts (1964).

These investigators were able to overcome both of the above

mentioned difficulties by assuming two stress axes, tensile

and compressive, which act parallel and perpendicular to

the rolling, direction, respectively.

By using the systems

of maximum resolved shear stress for a biaxial stress system

defined by the limitation that the tensile axis must be

perpendicular to the compression axis, prediction of a

[ lio] <112> texture for primary slip and a series of

other textures Including a [

1 1 2 \ <1 1 1 > for cross slip

was made. This analysis was experimentally confirmed by

the observed textures and by the appearance of cross slip

10 only in very dilute alloys and pure metals.

The qualitative evidence of correlation of texture

and cross slip was certainly significant since another

tool for investigating the plastic properties of metals

was made available.

The (llo\

<112> texture being a FCC alloy type is

generally referred to as the "brass type" while the {ll

2 J


111> texture being one of the textures reported for pure

FCC metals is generally referred to as the "copper type".

The pole figures obtained by the diffractometer

method can be described quantitatively, as well as quali­

tatively. Quantitative analyses of pole figures is possible

since the intensities of the x-ray peaks can be compared

with the intensity obtained from a sample of randomly

oriented grains. By using values of intensities obtained

from textured samples, which are multiples of the random

sample, it is possible to obtain values of intensity which

are reproducible.

However in order to obtain quantitative parameters

from the pole figures, for the purpose of reporting data

such as percent transition from one texture to another, or

for reporting the amounts of textures which are present in

a metal, it is necessary to use some relationship of the

intensities within the pole figure„

Smailman (

1 9 5 5 ) employed a convenient method of

stating the transition of one texture to another.


a {ill} pole figure for PCC metals, the ratio of inten­ sities


Io i ^ ( 2 5 or 30°) was reported as percentage transition.

I is the intensity

at alpha =

0 ° and 1 ( 2 5 or 3 0 °) ls intensity at alpha =

25 or 30°„ Liu and Richman (i

9 6 0 ) and Hu and Goodman (1 9 6 3 )

used a method essentially the same as Smallman*s except that

the ratio was taken of the peak height at the transverse

direction to the peak height at about

20 degrees from the

rolling direction (on the periphery of the {




A more quantitative treatment is offered by Bragg

and Packer (1964) in which the entire pole figure is con­

sidered through calculation of pole densities. The texture

is then represented by Gaussian or superposition of Gaussian

orientation distributions.

Dillamore, Smallman, and Roberts (1964) find that

the ratio of peak height measurement at the periphery used

by Liu and Richman (i

9 6 0 ) is a more sensitive intensity

ratio than that used by Smallman (1955). The intensity

peaks at about 20 degrees to the rolling direction remain


essentially stationary while the Intensity peaks at the

transverse direction move toward the center as the texture

changes from the [llOi <

112> (brass type) to the {5 3 1 \


112> (copper type).

By using the intensity ratio at the periphery.

Merlin! and Beck (1955) determined the copper-to-brass

texture transition and found that the main components of

the rolling texture gradually shift over a wide composition

range from pure copper to 70-30 brass. Hu and Goodman


1 9 6 3 ) have shown that the rolling texture of copper

gradually changes from the copper-type texture to the

brass-type texture as the temperature of deformation


A remarkable resemblance was found in the

rolling textures of copper deformed at -80, -140, and -I

9 6 0 C

to the room temperature rolling textures of brasses con­ taining

3 , 6 , and 10$ zinc, respectively, as reported by

Merlin! and Beck (1955).

Smaliman (1955) investigated the

texture dependence of a brass containing

5% zinc at different


At 200° C, the texture is the pure metal type,

at room temperature a 40$ transition to the brass type occurs,

while rolling at liquid air temperature produces almost a

1 0 0 $ transition to the brass type.

2.2 Stacking Faults

Stacking faults in metals, as the name implies, are

faults in stacking of atom layers . These faults cause a

region of higher energy compared to the usual lattice

stacking and are formed when dissociation of dislocations

13 occur.

The dissociation of dislocations occurs because

the energy of the dislocation is lowered by the dissocia­ tion process.

When the dislocation becomes dissociated,

or extended, it is split into two other dislocations called

partials which are of lower energy and repel each other.

The mutual repulsion of the partials creates an area in

which the stacking normal to the plane of the partials has

been altered from the usual FCC structure to a HOP structure.

The distance between the partials is a measure of the stack­

ing fault energy. The equilibrium distance between partials

is a result of two opposing forces: (a) the tendency of

the metal to be FCC rather than HCP, or the surface tension

of the fault, and (b) the repulsive force caused by inter­

action of the stress fields of the partials which tends to

separate the partials and thereby decrease the energy of

the stress fields.

Force due to surface tension repulsion force of partials distance

Sketch 1.

Surface tension and partial dislocation repulsion

vs. distance of separation of partials.


The distance at point p Is the equilibrium distance

between partials. For a metal of high stacking fault energy,

the slope of the surface tension is steeper and therefore

the stacking fault energy is higher.

The lower the stacking

fault energy, the greater the separation between the par-

tials and the wider the stacking fault.

The types of stacking faults which can occur in FCC

structures are due to ordering, quenching, climb by diffu­ sion, and plastic deformation.

Only the intrinsic fault

caused by plastic deformation and the twin fault caused by

diffusion are pertinent to this discussion. They will be

described here in more detail.

The stacking sequence perpendicular to the plane of

the fault can be compared schematically with the usual FCC

stacking as follows: usual FCC








A A A A A A A intrinsic fault





B B B B B B B B partial dis-

*"*C C C C 0 C S C location path







twin fault













In the deformation fault, the A layer above the

dislocation path has been changed to a B layer by the movement of atoms along a <112> direction so that the stacking has been changed from ABCABC. to ABCABCBCAB.„„„„

The twin fault is thought to form during recrystallization.

A dislocation which lies on the slip plane may

split up into two partial dislocations due to the energe­ tically favorable reaction

1/2 [lio] — > 1/6 [21l]

4- 1/6 [l21 _

where the terms in the equation indicate the magnitude and

direction of the Burger’s vector. The total dislocation is

said to have become extended. The relation of these vectors

on the close packed plane is shown below:


Sketch 2. Burger's vector relations on { 111

\ plane FCC


A marble model of splitting of the dislocation is

shown in Fig. 1 (a) and the resulting fault viewed perpen­

dicular to the close packed plane is shown in Fig. 1 (b)

and (c).

An annealing-twin fault is thought to form during

either recrystallization or during grain growth.



19AA) has proposed a theory of the generation of twins

from stacking faults during recrystallization.

Fullman and

Fisher (1951) have indicated that twins appear at grain

corners whenever the boundaries between a parent and its

neighboring grains are of higher energy than those between

the twin and these neighbors, the difference exceeding the

energy produced by the creation of a coherent twin boundary.

The coherent twin boundary is defined as the boundary at

which atoms of the twinned and parent crystal match up

17 perfectly across the twin boundary.

Boundaries at which

atoms do not match up perfectly are termed non-coherent

twin boundaries.

A coherent twin boundary in PCC metals has one

next-nearest violation, BOB, while a deformation fault

has two such violations, BOB and CBC.

The deformation

fault, therefore, is assumed to be made up of two twin

faults and consequently the stacking fault energy should

be twice the twin boundary energy (Read, 1953).

The measurement of stacking fault energy has yielded

a large discrepancy in values using either theoretical

calculation or experimental values.

Seeger, Berner and Wolf (1959) made measurements of

the temperature and strain rate dependence of the shear

stress corresponding to the transition from linear to

parabolic hardening in a single crystal stress-strain curve

to deduce stacking fault energies of several metals.


this treatment the onset of stage III in the single crystal

stress-strain curve is related to the stacking fault energy.

The results of this treatment yielded values of stacking

fault energies which are very much different from previously


An x-ray method which determines the. stacking fault

probability, of frequency, has been used by several inves­

tigators (Hu and Goodman, 1963; Wagner, 1957; Davies and

Cahn, 1962 j

Warren and Warekois, 1955, for instance). In


this method the peak shifts of certain planes ave measured

(usually the {lll\ to {

2 0 0 $ and | 2 2 0 j to {3 1 1 $

planes in FCC metals).

The peak shift is caused by lattice

strains due to cold work and small particle sizes as well

as by stacking faults. By application of Fourier analysis

to the displaced and broadened peaks it is possible to

determine the center of gravity and, therefore, the dis­

placement to a fairly good accuracy. The stacking fault

energy is related to the stacking fault probability as

given by the relation

3 alpha = (Nakajima, i

9 6 0 )

where alpha is the stacking fault probability and G, a,

p, E are the shear modulus, the lattice parameter, the

dislocation density, and the stacking fault energy, res­

pectively. In this method it is necessary to use a weighted

mean of the {111}

planes responsible for reflection. This

weighted mean becomes less appropriate for large stacking

fault probabilities (low stacking fault energy).

The most widely accepted values of stacking fault

energy are those derived from dislocation node measurements

by use of the transmission technique on the electron micro­ scope.

When three families of dislocations on the same

plane intersect, a triple node configuration results:


Sketch 3. Dislocation node containing faulted area.

Each dislocation is split into two of the three possible

mobile partial dislocations.

On half of the nodes, the

partial dislocations meet in such a way that the nodes can

open up into a triangle; on the other half of the nodes,

the necessary crossing over of the partials keeps the nodes

closed (Friedel, 1964). The stacking fault energy is then

calculated from the relation p

_ Gb L ij ~ 43.

6y where G is shear modulus b is Burgers vector

L is 4.55(2+v)/(1-v) for edge

dislocations and 5.P5(2-3v)/


1 -v) for screw dislocations

(v = Poisson's ratio)

y is inner radius of triangles

The equation is an approximate one because the core radius

of a dislocation is still uncertain and the variation of

dislocation energy along the curved length is difficult to



Another limitation to the dislocation node measure­

ment is that it is only applicable to metals of low stacking

fault energy.

As the stacking fault energy increases, y

becomes smaller until it reaches the limit of resolution

of measurement. The limit is reached at approximately


zinc in brass so that it is necessary to extrapolate to

obtain the value for pure copper. This extrapolation assumes

that the curve is continuous and follows the trend estab­ lished from

7 0 - 3 0 brass to 90-10 brass. It would be desirable,

therefore, to obtain stacking fault energies by another method

in the low zinc range to determine if the extrapolation which

is used is justified.

If the assumption is valid that the twin boundary

energy is one-half the stacking fault energy, then the area

of extrapolation could be confirmed or corrected.


three investigators have determined twin boundary energies

(Pullman, 1951; Bolling and Winegard, 1958; Inman and Kahn,

1 9 6 1 ). Only pure metals, copper and lead, were used in

these investigations, but it may be possible to extend the

method to alloys. The surface tension of a twin boundary

can be measured by using a mechanical analogy proposed by

Pullman (1951).

Consider a twin trace on a PCC crystal

whose surface coincides with the { liol plane.

Since PCC

metals twin on the

{ l l l i s

plane, then this twin plane will

be perpendicular to the surface.

The surface tension of

the twin boundary tends to pull the grain boundary in such

a way as to shorten the length of the twin boundary:


Grain A


Sketch 4. Twinned crystal

The surface tension of the twin boundary, then,

may be calculated by resolution of forces parallel to the

twin boundaries:



Sketch 5. Mechanical analogy of twin surface energy.



Resolving forces in the vertical direction yields: cosOg cos©^ - cos©^ cos




where F^ is the twin boundary energy and F^g is the grain

boundary energy.

Bolling and Winegard (1958) used the same

analysis for one-twin boundaries and employed the relation


©2 -P cos ©2 ).

It is seen from the above equation that to obtain

the stacking fault energy of a material it is necessary to

know the grain boundary energy.

In addition the angles

involved must be true dihedral angles or their orientation

must be known. The grain boundary energies for the solid

solution range of the Cu-Zn system have been calculated by

Taylor (1958). Barker and Parker (1945) have shown that if

a large number of measurements are taken, the modal value

should lie within 5° of the true value.

Bolling and Winegard


1 9 5 8 ) have used this latter assumption in determining the

interfacial free energy of twin boundaries in lead. Pullman


1 9 5 1 ) used the assumption that most of the grains were

oriented such that the { llo\ planes were parallel to the surface


2.3 Dependence of Texture on Stacking Fau1t Energy

The addition of a solute to a pure FCC metal has

been found to change the texture from the copper type to

the brass type (Merlin! and Beck, 1955; Smaliman, 1955).

Prediction of the type of texture obtained has been made

very satisfactorily (Dillamore and Roberts, 1964; Liu,

1964) on the basis of the cross-slip mechanism. The brass-

type texture is found to predominate in alloys where cross

slip is absent or where it occurs to a low extent; the

copper type is predominant where cross slip occurs to a

high extent.

The amount of cross slip, which changes the texture,

has been attributed to the stacking fault energy of the

material. The following schematic drawing illustrates why

cross slip is more difficult in metals with wide stacking

faults (low stacking fault energy): stacking fault partial dislocation stacking fault

Sketch 6 . Stacking fault with constriction


The extended screw dislocations which are on the slip plane

cannot cross slip unless they become constricted at some

point. More energy is required to constrict wide faults

than is required for narrow faults.

Seeger (1959) assumed

a model pile-up of stacking faults at Lomer-Cottrell locks.

The leading dislocation becomes constricted until cross

slip can occur. However, Thornton, Mitchell and Hirsch


1 9 6 2 ) suggest that cross slip can be activated by the

jogging of the glide dislocations with tangled dislocation,

dipoles, and prismatic loops as well as with Lomer-Cottrell

locks. Howie and Swann (

1 9 6 1 ) have indicated that the

addition of zinc to copper lowers the stacking fault energy

due mainly to the number of valence electrons in the alloys.

Swann and Nutting (

1 9 6 1 ) have suggested that lowering the

stacking fault energy by the addition of alloying elements,

reduces the size of the dislocation substructure cells and

thereby increases the dislocation density within the cells.

The dislocation density then, will increase as cross slip

decreases. The stacking fault energy, as determined by the

x-ray method, is inversely proportional to the dislocation

density so that the stacking fault energy should increase

with the amount of cross slip. Nakajima (i

9 6 0 ) attributes

the increase of stacking fault probability with additions of

solute to be due to the concentration of solute atoms around

faulted layers. This was verified by small angle scattering

experiments with Cu-Al alloys.


Since the texture is a function of the amount of

cross slip and the amount of cross slip is a function of

the stacking fault energy, then the texture should be a

function of the stacking fault energy.

If the stacking

fault energy can be shown to follow some exponential rela­

tion with solute additions and if the same relation exists

vary linearly with the stacking fault energy.

Smailman and Green (1964) have shown that in Ag-Au

alloys the texture changes in the same order as the stacking

fault energy change.

Dillamore, Smaliman, and Roberts (1964)

have shown that the amount of copper-type texture can be

used to classify pure PCC metals in order of their stacking

fault energy.

2.4 Effect of Solute Additions on Interfacial Energy

When a solute is added to a pure metal, the surface

tension of a metal surface, grain boundary, or twin boundary

is decreased. This decrease is due to segregation of the

solute atoms to either side of the boundary because these

boundaries act as a sink for the solute.

The influence of impurities, or solutes, on the

interfacial energies in metals has been studied extensively

from a thermodynamical approach. However, comparisons

between theory and experiment have not been made to any

great extent primarily because of the experimental difficulty

in determining the solute concentration at the interface.

The Gibbs adsorption isotherm gives a relation

between the change of surface tension of a liquid with

dilute concentration of solute dy/dC, and the excess con­ centration of solute in the surface, / :


/ " = -(C/RT) (dy/dC)„

This equation shows that, as a rule, solute elements which

decrease the surface tension avoid the surface.

It would

be expected that this relation might be applicable to solids

since solute atoms larger than the solvent atoms would

replace the latter and fill up gaps at the interface while

solute atoms smaller than the solvent atoms would help

relieve compressive stresses (McLean, 1957).

The observed results, in metals, are that solutes

lower the surface tension. Buttner, Funk, and Udin (1952)

have found that the surface tension of silver decreased by

a factor of approximately 3 by adsorption of oxygen. The

ratio of grain boundary energy due to surface energy

remained the same however. This is probably due to the

grain boundary energy decreasing at the same rate as the

surface tension because of pipe diffusion of the oxygen

along the grain boundary.

Bragg and Nye (19^7) found with the bubble model

that when bubbles larger or smaller than the average size

were present, a large proportion were situated at grain

boundaries. More evidence for solute segregation to grain

boundaries is given by the thermal etching phenomena. This

ridge-and-trough effect has been ascribed to the presence

of a second element partly because it differs from the

normal etching effect at a grain boundary, and also because

it varies in strength with composition and heat treatment.

This effect has been observed in several copper alloys

(Perryman, 1953) but not in high-purity copper.

Probably the most direct evidence obtained at present

for solute segregation to interfaces was obtained by use of

radioactive elements Introduced by Inman and Tipler (1958).

This technique involves breaking of alloyed test specimens

at a low temperature to obtain grain boundary fracture.

Successive layers of material are then removed by etching.

The etching solutions are analyzed for the radioactive

element. In this way Inman and Tipler (1958) found that

an addition of 0.5 w/o Pb to Cu reduced the surface energy

by a factor of approximately

2 . This results correlated

well with the Gibbs equation.


The objectives of this investigation were as

1. To determine the twin-boundary energy of brasses in the range of

0 to 12 percent zinc.

2. To analyze the twin-boundary energy method in an attempt to determine if the stacking fault energy is twice the twin boundary energy as presently supposed.

3. To seek a quantitative correlation between

stacking fault energy and rolling texture for brasses in

the low zinc range.



The brasses used in this investigation were prepared

from 99.999$ Gu, donated by American Smelting and Refining

Company, and 99". 999$ Zn, obtained from Consolidated Mining

and Smelting Company of Canada. The chemical analysis of

these materials is given in Table I.

To prepare the alloys, a master brass alloy of


1 6 .7 w/o Zn was prepared from the high-purity

Cu and Zn by melting the component metals in a sealed quartz

tube which was evacuated to approximately 10~° mm Hg„ The

molten brass was mixed by shaking the tube which was imme­

diately quenched into water. The resulting ingot, still

sealed in the quartz tube, was then homogenized for one

week at

8l6 °C. From this master brass alloy, specimens of

Cu-Zn alloys with zinc contents of approximately

1%, 2$,

5$, 7 $? 9$, 11$, and 13$ were prepared by diluting with

copper and processing similarly to the master alloy.

The alloys were then chemically analyzed for Zn

content by the x-ray fluorescence method using the XRD-5 GE

Spectrometer. The analysis consisted of the following

operations: (l) preparation of Cu and Zn liquid solutions of known concentration, (

2 ) preparation of standard brass

solutions by mixing the desired proportions of Cu and Zn

solutions, (

3 ) determination of the relation between



fluorescent Intensity of Cu and Zn/ (4) preparation of

solutions from Ingots, and (5) determination of Zn content

of Ingots by comparing intensities with standard curve„ A

detailed procedure for chemical analysis of alpha brasses

in the range of

Qffo to 15$ Zn is given in Appendix A„ The

alloys were analyzed for Zn content on both ends of each

ingot. The analysis indicated segregation to be within an

average of 0.14$ of each end with a maximum of 0.37$ dif­

ference occurring in one specimen. The*analyses of Zn

content for the alloys are shown in Table XI.

For the preparation of the pole figure specimens,

a sample from each ingot was rolled on a Stanat two-high

cold-rolling mill with 3-inch-diameter rolls.

The sheet

was reversed end-to-end for each pass to minimize asymmetric

pole figures. The samples were, rolled to

9 8 .3$ reduction

in thickness with no intermediate anneals, the sheets having

a final thickness of 0.007". The grain size of the ingot

before rolling was about

2 mm on the surface and cross-


This yielded approximately 40 grains per inch in

the transverse direction and

1 grain per inch in the rolling

direction of the rolled sheet.

Circular specimens were then

punched out of the sheets, the diameter coinciding with the

diameter of the x-ray specimen holder, 1-3/8". It was neces­ sary to reduce the specimen thickness to about

0 .0 0 3 ", giving

an x-ray absorption factor of about

2 , to insure adequate

intensity of x-ray beam penetration through* the specimens.


This was accomplished by chemical thinning in an etchant


30 $ nitric acid, 20$ acetic acid, and 50$ water.

Pole figures were determined with a GE-XRD-5

diffractometer equipped with a Proportional-counter and

an automatic pole-figure goniometer.

Cu K-alpha radiation,

and a Ni filter, were used in obtaining {lll^j pole figures

of the alloys„

A detailed procedure for obtaining pole

figures and the effects of the variables thereof is included

as Appendix B. In this investigation no random oriented

sample was used as an intensity basis since this author

has found that in a sample having pronounced texture, the

background intensity varies approximately linearly with

random sample Intensity.

The iso-intensity contours are

based on equal intensity increments at alpha =

0 (i.e. the

angle where the specimen plane bisects the angle between

the incident and transmitted beam).

For the measurements of twih-boundary energy, a

sample from each ingot was cross rolled on the Stanat

rolling mill to induce a £llo3 texture„ The samples

were rolled to 98 $ reduction in thickness with no inter­

mediate anneals. The final thickness of the sheets was

0.009". To obtain equilibrium annealing twins, strips

approximately 1/4" x

2 -1/2 " were cut from the cross-rolled

samples, and individually sealed in a partially evacuated

(50 mm Eg) 5/16" Vycor tubing. The specimens were then

broken out of the tubing and mounted in a casting resin

(no heat or pressure applied). The ends of the specimen

were bent up to provide a contact for electropolishing:

32 specimen end mount

Sketch 7. Specimen mount for electropolishing

The specimens were then ground lightly on 600 grit

paper and electropolished in an orthophosphoric acid

solution. Etching was done with a 2 to 1 mixture of

NHj^OH and by swabbing.

The angles between the twin traces and the grain

boundaries were measured at a large number of twin-grain

boundary intersections at a magnification of 500X using a

rotating mechanical stage on a Reichert metallograph. The

rotary stage has a calibrated degree scale with vernier so

that measurements can be made to 0.1°. The mean twin-grain

boundary energy and the standard deviation were calculated

for the copper and each of the brasses by the use of an

IBM 7072 computer.


5.1 Twin Boundary Energies

The frequency distributions of the apparent relative

energies calculated from the equation R = -(cosQ^ + cosGg),

where R is the ratio of twin boundary energy to grain boundary

energy, are plotted in Fig, 2, 3, and 4-. The distributions

appear close to normal and are all characterized by a slight

skew towards the high values of relative energy. However,

for a large number of observations the mean should be

normally distributed. The minimum number of observations

required to obtain a mean value characteristic of the

material was determined by plotting the mean and pange of

confidence against the number of observations as shown

in Fig, 5 for a brass containing 12,88 w/o zinc.

It can be

seen in the figure that at approximately

100 observations,

both the mean and range of confidence become essentially


It was assumed that this number of observations

would also apply to the rest of the alloys and the pure

copper. Due to time limitation and the discovery of an

annealing texture effect, to be discussed, the twin boundary

energies were not determined for the brasses containing

9.60 and

1 0 .8 8 weight percent zinc. Values of the relative

energies, along with the range of confidence are shown In


Table II. The range of confidence was determined for 90$

34 confidence limits from the relation u = x + t ^ q s

( x

) where

x is the mean, t 4 s

obtained from students’

statistical tables for the given number of observations,

and s(x) is the standard error. The standard error is

given by the ratio of the standard deviation to the square

root of the number of observations.

By using the values of grain boundary energies

calculated by Taylor (1958), the twin boundary energy was

obtained for each alloy. These are plotted in Pig.

6 .

5.2 Rolling Textures

Rolling textures obtained for the copper and brasses

in this investigation are shown in Pig. 7 and

8 with the

weight percent zinc as indicated. These pole figures are

in fairly good agreement with those previously reported by

Merlin! and Beck (1955) shown in Pig.

9 . The disagreement,

due to asymmetry, with previously reported pole figures: is

attributed to the large grain size present in the materials

used in this investigation.

The maxima shown near the center of the pole figures

shown by Merlin! and Beck (1955) were not plotted in Fig. 7


8 because these data were not used for texture deter­ mination.

Since no random sample was used, the pole figure

contours represent equal intensity increments determined

at alpha = zero.

The rolling texture of copper consists mainly of



the {53l} <001> which is close to the previously reported texture of -[ 123^ <4l2> by Hu and Goodman (

1 9 6 3 ). The rolling texture of the brass is (llO} <112> and coincides with the texture reported by Merlin! and Beck (1955)•

The data taken to indicate the copper-to-brass

texture transition were all derived from the periphery of

the pole figure where the ratio of the intensity of the

maxima at the transverse direction to the intensity maxima

about 20 degrees from the rolling direction was used. The

maxima close to the rolling direction remain essentially

constant while the maxima at the transverse direction

increase in Intensity with increasing zinc content. The

intensity maxima located near the center in each quadrant

in the pure copper correspond to some of the { 53ll <001>


These maxima merge into a single maximum, by zinc

additions, toward the transverse direction.

The intensity

at the transverse direction consequently increases.


measure of the amount of brass texture can therefore be

obtained by determining the Intensity ratio as described


This method of determining texture transition is

used by Dillamore, Smailman and Roberts (1964).

Intensity ratios obtained for the copper and brasses

are shown in Table III. The large grain size of the samples

which caused the asymmetry in the pole figures also affected

the intensity ratio but the general trend is an increase

36 with percent zinc. This trend, shown in Pig. 10, indicates that the brass texture increases very rapidly for the first additions of zinc to copper after which the ratio of texture change is low.. The rapid change in texture for low solute additions was also reported by Prois and Dimitrov (

1 9 6 1 )

-6 who found that copper additions as low as

15 x IQ- atomic

percent in aluminum caused a "change in texture.

The-asymmetry of the pole figures was confirmed to

be due to large grain size by comparing a pole-figure scan

of copper containing small grain size to a scan of a copper

specimen containing large grains„

The results shown in

Fig. 11 and 12 indicate that the small-grain copper has

greater symmetry. The specimen containing small grain

size was also of lower purity (

9 9 .99$) and indicated a

lower intensity ratio.

5.3 Dependence, of Texture on Stacking Fault Energy

Due to the inherent errors in the method used for

determining stacking fault energy which are to be discussed

and the error introduced by the large grain size in the

materials used in determining texture transition, only an

apparent trend in the texture dependence of stacking fault

energy can be given. Since no previous quantitative corre­

lation has been attempted for brasses, it is not possible

to compare results with values from the literature.


This approximate correlation is plotted in Fig. 14

where the values of stacking fault energy used represent

the best estimated values.

These values are given in Table

III. The value of intensity ratio for the 2.60 w/o zinc

and the value of stacking fault energy for the 5.30 w/o

zinc specimens were omitted from Fig. 14 because of their

large deviation from the other values due to the large

grain size in the former and an unknown reason in the



6.1 Twin Boundary Energies

An annealing-twin boundary possesses an energy which

is higher than the average energy of the surrounding lattice.

This is confirmed by the observed tendency of the twin

boundary to shorten itself by pulling at the grain bound­

ary and also by the well-known preferential etching at these


Values obtained for twin boundary energies of pure

metals depend on several variables. These probably include

(a) purity of the metal, (b) annealing atmosphere, (c)

annealing temperature and time, (d) cooling rate from the

annealing treatment, (e) values of grain boundary energies,

(f) the angles measured at twin-grain intersections and

(g) effects of annealing texture.

In general, the twin

boundary energy is expected to be increased by higher purity,

a decrease in contaminant in heat-treating atmospheres, a

lower annealing temperature or insufficient time at tempera­

ture, a slower cooling rate from the annealing treatment,

and increasing deviation of the annealing texture from a


type. The remaining two variables, values of grain

boundary energy and measured angles, will increase or decrease

the twin boundary energy depending on existing conditions.


6.1.1 Purity of Material and Annealing Atmosphere


Purity of material and annealing atmosphere affect

the twin boundary energy in a similar manner to that of

solute additions. Discussion of this effect is given in a

later section. •

6,1.2 Annealing Temperature and Time

When a thin sheet of polycrystalline metal is

annealed at a high temperature for a long period of time,

the grain boundaries will migrate so as to reduce their

line tension (Barker and Parker, 1945). This results in

having, most of the grain boundaries lying perpendicular to

the axis of the sheet.

If equilibrium is not reached and

the grain boundaries lie at some angle not perpendicular

to the sheet, the apparent grain boundary energy will be

lower than its equilibrium value. This would have the

effect of raising the measured value of R, the ratio of

twin boundary to grain boundary energy, thereby yielding

a higher value of twin boundary energy.

By comparing the value of R, 0.076, for copper

obtained for an annealing temperature of 7150C, in this

investigation, with that obtained by Pullman (1951),

0.045, for an annealing temperature of 950°C, it appears

that the annealing temperature effect is present as


discussed above„ However, this assumes that all other

factors are equal which is not necessarily the case.

6.1.3 Pooling Rate

If the metal is quenched from the annealing heat

treatment temperature, the twin and grain boundary energies

are both lowered. The energies are not necessarily lowered

equally, the lowering of the grain boundary energy probably

being more severe. This lowering of energy is due to

impurity diffusion to the boundaries (Inman and Tipler,

1 9 5 8 ). The general effect of this variable is to appear

to increase R when the metal is quenched. Unfortunately,

the cooling rate was not given by the previous investiga­ tors (Pullman, 1951; Inman and Khan, 1951) so that compari­

son of results on this basis is not possible. However,

cooling rate might not affect R to any great extent since

Flinn (i

9 6 0 ) has reported that cooling rate varying over

a wide range of temperatures does not affect the solid

solution hardening.

6.1.4 Grain Boundary Energies

The experimental values reported for grain boundary

2 energy of copper vary from 490 ergs/cm (Pullman, 1951) to


640 ergs/cm (Bailey and Watkins, 1950). The discrepancy

of these values could possibly be due to the purity of the

copper. Bailey and Watkins (1950) used a higher purity

copper, 99.998$, compared to Pullman (1951) who used OFHC

copper which is of

9 9 .98$ purity.

The grain boundary energies derived by Taylor


1 9 5 8 ) and plotted in Pig. 6 , were obtained by using the p value of

58 0 ergs/cm for the grain boundary energy of

copper. This has the effect of giving a lower value of

twin boundary energy if it is assumed that 640 ergs/cm

is a preferred value.


6.1.5 Angles Measured at Twln-Grain Intersections

Although the energy of the twin boundary is higher

than the surrounding lattice, its formation is a result

of lowering of energy. The most accepted mechanism of

twin formation is that given by Pullman and Fisher (1951)

in which they propose that annealing twins form at a moving

three-grain junction.

As the junction moves, by the process

of grain growth, the orientation across such a junction can

reach the orientation required for twinning.

Since the

twin boundary energy is lower than the grain boundary energy,

the twin formation is energetically favorable. However,

this hypothesis does not explain the almost universal

occurrence of twins in pairs. This twin-pair occurrence

might be explained as follows:

Consider an annealing twin being formed at a three

grain junction:


8 . Formation of annealing twin.

The energy across grain boundaries is considered

to be constant except for sraall-angle grain boundaries

where the energy is less. This has been calculated theo­

retically by Read and Shockley (1950) and agrees well with

experimental measurements.

The energy across AT, however,

is not equal to the energy across AB since the boundary AT

was not formed by exactly the same mechanism.


to Pullman and Fisher (1951), the energy across AT is

lower than the energy across AB since observations of

three-grain junctions show the angle BC is less than the

1 2 0 ° angle observed when the three grains are of equal

energy. However, according to Smith (1950) this angle is

reached when equilibrium is attained and grain growth


Annealing twins would then not form from 120°

grain junctions since no grain boundary movement occurs.

If it is assumed that the energy across AT is actually

greater than the usual grain boundary energy, then it is

possible to give a mechanism for twin-pair occurrence.

The increasing energy across AT and CT, as the grain B

moves toward C, would tend to inhibit the grain boundary

movement as the energy along AT approaches the energy for


grain boundary movement. This movement could resume if

another twin boundary was formed, thereby causing the

grain B to revert to its lower energy orientation. The

result of this mechanism would be for twins to tend to

occur in pairs which agrees with observations.

From the above discussion, the width of the twins

would be a function of the orientation of T with respect

to grains A and C. The energy across AT, as well as across

TC, would be expected to vary depending on the twin orienta­

tion and would be lower for some orientations of T. The

observed width would then be a result of an equilibrium

situation, the energy due to twin width being an interme­ diate in energy as that across AT and across CT. Observa­

tion of twinning, in this investigation, shows at least

two other cases:


Sketch 9

Some observations of twinned grains

44 than the energy CT. In the second case (b), the difference

in width of twins is due to the difference in orientation

of A and A', T and T' probably being close to the same

orientation. The fact that coherent annealing-twin bounda­

ries do not move perpendicular to the twinning plane has

been shown by Pullman (1951). The non-coherent twin

boundaries, however, do migrate parallel to the coherent

twin boundaries (Pullman, 1951).

It is possible, therefore,

for the twin in case (a) to become of constant width if

grain boundary movement causes the twin to extend from

grain C to grain D and if the orientation of grain D is close to that of grain C.

The significance of the above discussion of occur­

rence of twin pairs is that a source of error in the method

of computing twin boundary energies is inherent in the

method used by this investigator and all previous ones

(Pullman, 1951J Bolling and Winegard, 1958j Inman and


1 9 6 1 ). In this method the force exerted on the twin

boundary by the twin-to-grain boundary AT is assumed to be

the usual grain boundary force while it is probably different

from this value„z The error is present whether it is assumed

that the AT boundary is of lower energy (Pullman and Pisher,

1 9 5 1 ) or higher energy as suggested here.

Since measurement of the energy of the AT boundary

would be difficult, a better method would be to use the

45 relation

R = -(cosQ^ + eos©^) for determining the twin boundary energy.

©^ and ©^ are

the angles on either side of the twin pair.

This method

would eliminate the use of the AT boundary energy.


would also probably eliminate the negative values of R

which are common using the present method. It is incon­

ceivable to expect that the twin boundary energy is greater

than the grain boundary energy as is implied by the negative

values of R. This investigator did not observe any case in

which the sum of the angles on either side of the twin pair

was less than l

8 0 ° during a random check of about 3 © twins.

6 .1 .6

Effects of Annealing Texture

When the angles between a twin trace and the grain

boundary on both sides of the twin are measured, the

measured value of R is in general higher than the true

value depending on the angle of the twin surface with the

perpendicular to the sheet surface. The larger the angle,

the higher is the measured value of R. In the special

case where the {liol plane coincides with the specimen

surface, the ^lll] twin planes will lie either at

0 ° or

54.7° to the surface perpendicular„ Four of the {ill]

planes in the unit cell will lie at

0 ° while the other

four will lie at 54.7°. Since, in compression rolling

(viz. cross rolling), the texture of the rolling sheet is


predominantly of the tliol type, Pullman (1951) assumed

that by measuring only the twins with boundaries nearly

perpendicular to the surface, the values of R obtained

would be the true values. However, during the necessary

annealing treatment after rolling, the texture is changed

to a pronounced cube texture, {ooijj <100> (Merlin! and


1955) for instance). With this type of texture,

all the

{ 111 ^ planes in the unit cell make an angle of

35.3° with the surface perpendicular. The values obtained

by Pullman, then, appear to have an inherent error due to

his assumption. Although he stated that the orientation

of the twinned grains was determined, the method used,

that of measuring angles between slip traces, does not

yield unique solutions and in fact may yield erroneous

solutions (Cullity, private communication). To determine

grain orientation by means of slip and twin traces, four

octahedral traces are required (Takeuchi, 1959). This has


proved unattainable by this investigator, on copper and

brasses, regardless.of the amount of deformation introduced.

The same assumption was used by Bolling and Winegard


1 9 5 8 ) in their investigation of annealing twins in lead.

Although the annealing texture of lead might yield a pro­

nounced texture in which both the twins and grains are

perpendicular to the sheet surface, the occurrence of this

seems unlikely. Inman and Khan (

1 9 6 1 ) circumvent the

texture limitation by use of transmission-electron micro­ scopy.

In this method, electron-diffraction patterns of

the twinned crystals are used to determine the orientation

of each twinned grain in which the twin angles have been


In this investigation, it was assumed that a

pronounced cube texture was present in the copper and

brass specimens. For the copper, this was confirmed by

a pole-figure scan of a specimen that had been annealed

at the same temperature and for the same time as the

specimen from which the twin angles were measured.


the brasses, however, the assumption probably leads to

erroneous values of R, since Merlin! and Beck (1955) find

that the decrease in the amount of cube texture is very

great between

0 and 3^ zinc, and it is also quite consider­ able between

3: and 6$ zinc, while in a 10$ zinc brass,

the amount of cube texture is very low. Due to the radical

change in annealing texture in these low-zinc brasses as

shown by Merlin! and Beck (1955), it would be difficult

to make the necessary correction of R. The values plotted

in Figure

6 for R and twin boundary energies are the values

that were measured, without correcting for the texture

effect. However, since the cube texture decreases with

percent zinc, it is reasonable to assume that the angle

of the {llli

twin traces with the surface perpendicular

increases as the high index annealing texture components


(e.g. \_214.!, [358; ), reported by Merlin! and Beck (1955),

increase. This has the effect of increasing the measured

value of R. Pig.

6 should then actually show a decrease

of R with percent zinc rather than the approximately con­ stant value of R.

For the copper, the value of R could be corrected

by use of the relation

R = R' cos 35.3° = O.

8 1 6 R 1,

where R' is the measured value and 35.3° is the angle

between all the { m l twin planes and the surface per­ pendicular in the case of a pure cube texture.

This yields

a corrected value of R of O.O

6 1 8 for a twin boundary energy p of 35.8 ergs/cm as opposed to the measured R of

0 .0 7 6 for.


a twin boundary energy of 43.5 ergs/cm , using the value of

580 ergs/cm for the grain boundary energy as given by

Taylor (1958).

6 . 2

Effect of Solute Addition on Twin Boundary Energy

The effect of a solute addition is to lower the

twin boundary energy due to segregation of solute atoms

to either side of the boundary. The largest effect occurs

at the first additions up to about

1 atomic percent zinc

in brass as indicated in Figure

6 . In this range, R drops

by a factor of about 1.8. This expected large decrease

at low concentration -is supported by the results of McLean,


Inman, and TIpler (

1 9 6 1 ), who reported an energy reduction

at the grain boundaries of copper when

1 weight percent Pb

was added.

The most obvious mechanism for causing the large

decrease in surface tension is that of saturation of solutes

at the interface.

If an estimate of the number of solute

atoms necessary to saturate each atom plane of every dis­

location, grain boundary, and twin boundary is calculated,

the total impurity content necessary for saturation will be rather low. For example, in a cubic centimeter block of



copper having

10 dislocation lines per cm , a grain size of

0 .0 6 mm, and 5 twins per grain, the atomic percent of solute needed to produce a monatomic solute layer on each interfacial plane and dislocation line is atoms/dislocation line+atoms/grain boundary+atoms/twin boundary atoms/cm^ ' which yields about

4 x 10

15 + 160 x 1015 + 480 x IQ15 _ 644 x IO15

6 x 10 2 2 6 x 1 0 22 or approximately 1 .0 7 x 1 0 “^ atomic percent.

Although the curve of R vs w/o zinc shown in Figure


indicates a knee at

1 w/o zinc, the actual knee could be at

lower values of zinc. It would seem, therefore, that the

twin boundary energy decreases until solute saturation is


reached after which the energy decreases at a slower rate

due to other mechanisms such as elastic interaction of

solutes with the solvent lattice and electronic disturbances.

With reference to the above discussion, the relative

importance of these effects may be estimated as follows:


Purity increases the value of R by about a

factor of

2 from OEHC (9 9 =.98%) copper to 99.9990 copper,

2 .

Annealing atmosphere decreases R as a function

of the impurity concentration of the surroundings by prob­

ably only a slight amount, providing the atmosphere is

relatively clean.

3. Annealing temperature and time increase R

but not to any great extent for any values obtained since

the temperature and time are considered adequate for forma­ tion of equilibrium twins,

4., Cooling rate from the annealing treatment decreases R to a small extent if a slow rate is used.


The suggested correction on the method of

measuring R affects the mean value of R by an unknown


6 .

The annealing-texture effect yields values which are too high by a factor of

0 .8l6 when it is assumed

that a ^lio] texture predominates. This effect probably

becomes more serious for brasses.

The purity, method of measuring R, and annealing

texture are therefore considered the principal factors

which contribute to variation in the values of R.


Prom the above discussion, it appears that the true

values of twin boundary energy for copper and brasses remain

to be determined. The method used by Inman and Khan (

1 9 6 1 )

is probably the best approach to this type of determination.

The optimum values for R could be determined by measuring

twin angles and determining the twinned grain orientation

by use of transmission-electron microscopy and utilizing

the suggested correction of choosing only the exterior

angles of the twins.

The present value of twin boundary ratio,

0 .0 2 , for

OFHC copper as determined by Inman and Khan (

1 9 6 l) is prob­

ably too low while the value determined by Pullman (1951)

of 0.045 is probably too high. The true value probably

lies within these two values. A rough estimate is 0.03

for OFHC copper. For high-purity copper, the value is

about twice the value for OFHC copper.

6.3 Twin Boundary Energy vs Stacking Fault Energy

The twin boundary energy is a measure of the energy

associated with the free energy of the twin fault surface.

This free energy is also proportional to the radii of

curvature at dislocation nodes, where it is assumed that

the repulsion of the partial dislocations is equal to the

free energy of the stacking fault (Friedel, 1964).


6.Sol Present Assumption

The tension, or free energy, per unit area of a

stacking fault is usually assumed to be twice the twin

boundary energy. The basis for this assumption is that

the stacking fault is considered as a pair of parallel and

adjacent twin planes B and C defining a twin band of minimum








C -













B —

0 —~

The difference between.the twin fault energy and c .


the deformation fault energy as determined by measurements

of twin angles and dislocation-node measurements, res­ pectively, probably includes the following factors: (a)

temperature effect, (b) atom neighbors effect, and (c)

electronic effect.

6.3,2 Temperature Effect

The deformation fault energies are usually measured

under conditions existing at room temperature while the

twin fault energies are measured for conditions existing

at the temperature of annealing.

If the stacking fault

energy decreases with temperature, as predicted by the

Gibbs equation, AF = AH - TAS, then the twin fault energy


should be lower than the stacking fault energy. At present,

hardly any information is available on the effects of

temperature on stacking fault energy except by the texture

dependence, in which a decrease in stacking fault energy

with a decrease in temperature is indicated.

6.3.3 Atom-neighbors Effect

One approach to the effect of the violation of

neighboring atom stacking due to faulting would be to

consider the change in distance of like atoms and obtain

some value akin to the configurational entropy.

In all three cases; usual stacking, twin fault

stacking, and deformation fault stacking, the distance

between nearest neighbors is the same. The distance

between next-nearest neighbors, however, is different

between the two types of faults.

In the deformation fault,

the B atoms are closer together than in the usual stacking

by a factor of 2/3,-the A. atoms are farther apart by a

factor of 3/5, and the C atoms are closer together by a

factor of 2/3.

In the twin fault, the B atoms are closer

together by a factor of

2/3 , the A atoms are farther apart

by a factor of 3/4 and the C atom distance remains unchanged.

Strictly on the basis of the above difference

between a twin and a deformation fault, the fault energy

should differ by a factor obtained from the A and C atom


separation differences. The ratio of like-atom separation

factor between a stacking fault and a twin fault is then the three factors being related to the

A, B, C atom separa­ tion distance, respectively.

The products of the individual factors rather than

the sum have been used here only on an intuitive basis.

The entropies of a system are usually additive but these

factors are intimately interrelated with each other.


addition, the factor for the B atoms remains unchanged and

a summation of factors would cause an additive term of

unity to be introduced.

The only other approach to this type of analysis

seems to be that of using the entropy relation;

S = k In W,

M i where k is the Boltzman constant and W = ^ T"n j > h* e

However, the variables for the factorial terms would be

difficult to choose to fit this system.

6.3,4 Electronic Effect

■ -

Seeger (1957) has stated that as a result of next-

nearest neighbor violations, the energy of the electron

gas (Fermi energy, Ef ) changes due to the change in wave

vectors near a Brillouin zone boundarye This change must


be such that the introduction of a stacking fault results

in an increase of energy.

For a twin fault, the difference in Fermi energy

for the copper matrix and the twin boundary should be pro­

portional to the twin boundary energy. For a deformation

fault, which consists of two twin faults, the Fermi energy

increase at the fault should be due to a superposition of

the two twin faults as shown in Sketch

1 0 , The Fermi

energy is shown as a continuous function across the fault

but having a maximum at the fault, A rapid decrease on

each side of the fault is shown to indicate the tendency

for the Fermi energy to be constant.

Graphical superposition of the amplitudes of

for the individual twins yields a factor of about

1 ,7 5 for

the electronic effect. This superposition indicates that

the deformation fault energy is

1 ,7 5 times greater than

the twin boundary energy.

Since the conductivity is decreased by the presence

of the twin boundary because of the increase in E^, the

magnitude of the change in Ef should be indicated by

electrical resistivity measurements.

Indications that

the conductivity does in fact drop across twin boundaries

have been obtained by Hancock, Keating, and Murphy (1963)-


6.3=5 Summation

Neglecting the unknown temperature effect and using

the suggested approximate value for OEHC copper (p. 51) the

stacking fault energy as determined from twin boundary

measurements is approximately


0 .0 3 )(5 8 0 )(1 .2 )(1 .7 5 ) = 36.5 ergs/cm2 .

For high purity copper this becomes about twice as much

or about 73 ergs/cm2 which agrees with the value obtained

in this investigation of


6 2 )(5 8 0 )(1 .2 )(1 .7 5 ) = 75= 6 ergs/cm2 .

It is possible that the error from using the previously

mentioned angles in. the twinned crystal will affect this


The estimated values of stacking fault energy

determined using all the corrections described are shown

in Table III along with the values obtained by dislocation

node measurements. The values by the latter method were

obtained from Smaliman and Green (1964) who used the Siems

correction (

1 9 6 1 ) on the data obtained by Howie and Swann


9 6 1 ) on 99=999$ purity copper and zinc. The values

obtained by the two methods are plotted in Fig. 13 where

it is seen that the comparison is quite satisfactory at

the overlap region of the two methods. It is believed by





Sketch 10. Schematic plot of Fermi energy vs 111 d-spacing.

this author that the values between

0 and 10 percent zinc

are given more accurately by the method involving twin

boundary energies.

If the factors involving the accuracy of twin

boundary energies are considered for the brasses, as is

possible by use of the electron-transmission-microscopy-

technique of determining the orientation of the twinned

grains, it should be possible to obtain accurate and

reliable data for stacking fault energies.


6.4 Texture Intensity Ratios

The intensity ratio used in determining texture

transition is independent of the experimental conditions

such as beam slits or tube power, and absorption factor.

It is dependent on the (a) percent reduction, (b) grain

size, and (c) purity of the specimen.

Braybrook and Calnan (1957) have shown that at

high reductions, abbut

97 to 990, the amount of texture

increases very rapidly with increasing reduction. Conse­

quently, a very small difference in percent reduction would

affect the intensity ratio.„ In this investigation, the

ingots were in the form of cylinders before the rolling

operation so that some error in the percent reduction

could have been introduced. The brass specimen containing

2 .6 0 w/o zinc deviated by a large amount from the other

values of intensity ratio due to this reason or due to its

large grain size.

Grain size has been reported to have little or no

effect on the intensity ratio by Dillamore, Smaliman, and

Roberts (1964), for the range of grain size from 10 g/mm

2 p

to 1500 g/mm «• The grain size for the materials used in

this investigation was approximately 2 g/mm . Unless the

grain size causes very much asymmetry the intensity ratio

should not be significantly changed„ The 2,60 percent

zinc specimen, however, had a very large grain size,

approximately 2 g/ojn . This large grain size is probably

the principal factor which caused such a large deviation

of the intensity ratio in that specimen. The values for

the other specimens were probably also affected by the

grain size although to a lesser extent,

The effect of purity on intensity ratio can be

seen in Fig, 11 and

1 2 , The 99»99$ copper yielded an

intensity ratio which is greater than the 99.9990 copper

by a factor of approximately

2 ,

6,5 Effects of Stacking Fault Energy on Rolling Textures

The analysis of Dillamore and Roberts (1964) has

shown that the brass-type texture is due to primary and

conjugate slip while cross slip produces a copper type


Seeger (1957) has attributed the ease with which

cross slip can occur, at a given temperature, to the width

of separation of partial dislocations and hence to the

stacking fault energy. The fundamental factor governing


the change in rolling texture from the copper to the brass

type would then be expected to be the decrease in stacking

fault energy„

6.5.1 Rolling Texture vs Cross Slip Mechanism

In analyzing the texture development, the assumption

of Boas and Hargreaves (1948) that deformation mechanisms

of individual grains can be considered to be closely related

to those operating in a single crystal, has simplified the

prediction of textures. Using this assumption, Dillamore

and Roberts (1964) have shown that the absence of cross

slip results in a pure brass texture. Employing the method

of brass texture component in a brass specimen by using

the intensity ratio described earlier, an indication of

the amount of cross slip occurring can be obtained.


the intensity ratio decreases, the amount of cross slip


To cause cross slip to occur in alpha brass of

high zinc content, it would be necessary to apply a very

high stress because of the high stability of the brass

texture. This high stability has been shown by Dillamore

and Roberts (1964) to be due to the fact that a large rota­ tion of the stress axes would have to occur.

The results shown in Pig. 10 indicate a amount of cross slip for the brass containing 1 2 .8 8 w/o zinc having an intensity ratio of 1 . 2 and a large amount


of cross slip for the pure copper with an intensity ratio

of 0.2. Furthermore, the figures indicate that the differ­

ence in amount of cross slip occurring is greater between

0 and 1 w/o zinc than it is between 1 w/o and 12 w/o zinc.

This would seem to indicate that the impurity content in

pure metals is a major factor in affecting cross slip.

6.5.2 Cross Slip Mechanism vs Stacking Fault Energy

■When a screw dislocation moves across the slip plane

it may encounter an obstacle such as an inclusion, dipoles,

prismatic loops, or dislocation entanglements. These have

the effect of impeding slip thereby resulting in dislocation

pile ups. If the piled up dislocations are not extended it

is possible for them to loop around the obstacle and resume

their movement by leaving a looped portion around the obsta­

cle. If the dislocation can move to another slip plane, or

cross slip, then this will occur since this produces a path

of lower resistance. On the other hand, if the dislocations

are extended, cross slip becomes more difficult because the

partial dislocations must form a constriction before cross

slip can occur. The wider the separation between partials,

the greater will be the stress required to constrict the

fault. This indicates that a metal with wide faults, or low

stacking fault energy, will not cross slip as readily as a

metal with narrow faults, or high stacking fault energy.


The results shown in Pig. 13 suggest that adding

zinc to copper lowers the stacking fault energy indicating

a decrease in amount of cross slip with zinc content. The

character of decrease as shown by this investigation

emphasizes that the amount of cross slip which occurs is

very sensitive to the initial solute additions.

6.5.3 Rolling Texture vs Stacking Fault Energy

Since both the stacking fault energy and the amount

of copper texture decrease in a similar manner with zinc

additions to copper it may be postulated that the change

in rolling texture is due to a change in stacking fault


Pig. 14 shows that the decrease of intensity ratio

with stacking fault energy is not linear in the range of

brasses from 0 to

1 2 .8 8 w/o zinc. However, the discre­

pancies in both the values of intensity ratios and stacking

fault energies could affect this correlation.

A linear

correlation would not necessarily be expected, even if the

amount of cross slip were a linear function of stacking

fault energy, because the intensity ratio could be an

exponential function of the amount of cross slip. The

lack of linearity in the logarithmic plot shown in Pig. 14

could be related to the discrepancies in the experimental


It would be naive to attempt a mathematical corre­

lation using the experimental values obtained in this

investigation but it is indicated that the rolling texture

is definitely a function of stacking fault energy.


Twin boundary energies were determined for high-

purity copper and high purity brasses in the range of


weight percent to

1 2 .8 8 weight percent zinc. The factors

involved in the accuracy of the values obtained in this

and previous investigations were considered.. These factors

are (a) purity of material, (b) annealing atmosphere, tem­ perature, and time, (c) cooling rate from annealing treat­

ment, (d) values of grain boundary energies, (e) angles

selected for measuring, and (f) effects of annealing


Of these, (a), (d), and (e) are considered the

most important.

The effect of zinc additions to copper was found

to lower the twin boundary energy at a rapid rate for the

initial zinc additions„ After approximately

1 w/o zinc

addition, the twin boundary energy remains essentially

constant or decreases at a.slow rate.

Analysis showed that the stacking fault energy is

related to the twin boundary energy by a factor of about

2 .1 which agrees closely with the present assumption, in

the literature, that the stacking fault energy is twice

the twin boundary energy. The stacking fault energies

found in this investigation agree with values determined


by the method of measuring radius of curvature of disloca­


tion nodes. The values disagree, however, with those in

the extrapolated region of the latter method.

Rolling textures for copper and brasses were deter­ mined by pole figure analysis and were found to agree gene­

rally with previously reported ones. However, an asymmetry

in the pole figures was present in the results of this

investigation. This asymmetry was attributed to the large

grain size present in the rolled condition.

A quantitative parameter was obtained from the pole

figures by using the ratio of intensity at the transverse

direction to the intensity approximately

2 0 ° to the rolling

direction. This parameter was plotted against values

obtained for stacking fault energies and a trend was

obtained which indicated that the rolling texture is a

The values of stacking fault energy determined in

this investigation could be ameliorated by eliminating the

measurement of interior angles in twins through the use of

the electron transmission microscope technique to measure

the angles and determine orientations of the twinned grains.

In addition if the grain boundary energies for the brasses

could be experimentally determined, the accuracy of values

for stacking fault energy would be further improved.


Prom the results obtained In this investigation,

the following conclusions have been drawn:

8 .1 Twin Boundary Energies

It is possible to determine twin boundary energies

for copper and alpha brasses using the method involving

measurement of twin boundary dihedral angles.

This method, as used in this investigation, as

well as by Inman and Khan (

1 9 6 1 ), Pullman (1951), and

Bolling and Winegard (1958) contains an error in assuming

that the grain boundary energy is equal to the twin-to-

grain boundary energy. The latter authors -- Pullman,

Bolling and Winegard — are believed to have an additional

error in their data* The former assumed a fllQ}


was present while it is fairly well known that the copper

annealing texture is of the {001}

type. The latter

assumed that the dihedral angles measured were true angles.

8 .2 Stacking Fault Energies

The twin boundary energies determined by the above

method can be used to obtain the stacking fault energy

contrary to the conclusion of Thornton, Mitchell, and


Hirsoh (

19 6 2 )0 The twin boundary energy was found to be

related to the stacking fault energy by a factor which is


2 „I,

The purity of the copper affects the stacking

fault energy significantly. This may account for some

of the variations in this value among several authors„

Experimental values of stacking fault energies

obtained in this investigation agree well with corres­

ponding values obtained from corrected dislocation-node

measurements by Howie and Swann (

1 9 6 1 )0 They disagree

with those in the extrapolated region of the latter method

The values obtained in this investigation for this region

are believed to be more reliable than those obtained by


8.3 Effect of Stacking Fault Energy on Rolling Texture

The rolling texture has been found to be dependent

on the stacking fault energy for the range of brasses

between 0 and

1 2 ,8 8 w/o zinc. Discrepancies present in

both experimentally determined parameters preclude a

mathematical correlation.


Bailey, (L L e J „ , Watkins, H. C., Proe. Phys. Soc, B, v

0 63,

P» 350, 1950

Boas, W„, Schmid, E„, Z„ Tech, Physik, v. 12, p„ 71,


Boas, W , , Hargreaves, M, E,, Proc, Roy, Soc,, V, 193, p,

9 6 ,


Bolling, G, P., Winegard, W. C„, Jour, Inst. Metals, v.

8 6 ,

p. 492, 1958

Bragg, R. H . , Packer, C„ M., Jour, Appl. Phy,, v, 35,


1 3 2 2 , 1964

Bragg, W. L„, Nye, J. F,, Proc. Roy, Soc., v.

1 9 0 , p. 477,


Braybrook, R. F., Calnan, E. A., Jour, Inst. Metals,


8 5 , p, 1 1 , 1956

Brick, R, M., Cold Working of Metals, ASM, Cleveland,

Ohio, p. 99, 1949

Buttner, F. H., Funk, E. R., Udin, H., Jour. Phy. Chem.,


5 6 , p. 657, 1952

Calnan, E. A., Clews, C. J. B . , Phil. Mag., v. 41, p.

1 0 8 5 ,


Dillamore, I. L., Roberts, W. T., Acta Met., v. 12, p.

2 8 1 ,


Dillamore, I. L,, Smaliman, R, E . , W, T., Phil,

Mag., v.

9 , p. 517, 1964

Davies, R. G., Cahn, R. W,, Acta Met., v. 10, p.

6 2 1 ,


Flinn, P. A,, Strengthening Mechanisms in Solids, ASM,

Cleveland, Ohio, i 960


Priedelj, J., Dislocations, Addison-Wesley Publishing Co.,

Reading, Massachusetts, 1964

Frols, C., Dimitrov, 0., Comptes Rendes, T

2 5 2 , No. 10,

p. 1465, 1961

Fullman, R. L., Jour. Appl. Phy., v. 22, p. 448, I

9 5 I

Pullman, R. L., Fisher, J. C., Jour. Appl. Phy., v. 22,

p. 1350, 1951

Hancock, J, R., Keating, K. L., Murphy, D. J., AEG Report,

Dept. Met. Engr., Univ. Ariz.,

1 9 6 3

Barker, D., Parker, E., Trans. ASM, v. 34, p.

1 5 6 , 1945

Howie, A., Swann, P. R., Phil. Mag., v. 6, p. 1215,


Hu, B . , Goodman, S. R., AIME Trans., v„ 227, p.

6 2 7 ,


AIME Trans., v. 194, p.

7 6 , 1964

Inman, M. 0., Khan, A. R., Phil. Mag., v. 6, 1 9 6 1

Inman, M. C., Tipler, H. R., Acta Met., v. 6, p. 73*


Kittel, C., Introduction to Solid State Physics, John

Wiley and Sons, New York, 1956

Liu, Y. C., Richman, R. 2 1 8 , p. 6 8 9 , i 960

Mathewson, C. H . , Trans. ASM, y. 32, p.

3 8 , 1944

McLean, D., Grain Boundaries in Metals, Oxford, 1957

Merlini, A,., Beck, P. A., AIME Trans., v. 203, p.

3 8 5 ,


Nakajima, K., The Research Inst, for Iron, Steel, and

Other Metals, Tohoku University, i 960

Perryman, E. C. W., J. Metals, v . 5, p. 906, 1953

Pickus, M. R., Mathewson, C. H . , Jour. Inst. Metals,

v. 64, p. 237,


Read, W. T., Jr., Dislocations in Crystals, McGraw-Hill

Book Co., Sew York, 1953

Read, W. T., Shockley, W., Phys. Rev., v.

7 8 , p. 2 7 5 ,


Seeger, A., Dislocations and Mechanical Properties of

Crystals, John Wiley, Hew York, 1957 —

Seeger, A., Berner, R., Wolf, H „ , Z. Physik, v. 155, P.


Siems, R., Delavignette. P., Amelinckx, S., Z. Phys.,

v. 165, p. 502, 1961

Sraallman, R. E., Jour. Inst. Metals, v. 84, p. 10,


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Research Inst., Ser. A p v. 11, p. 8l, 1959

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Taylor, J. W., Jour. Inst. Metals, v. 86, p.

4 5 6 , 1958

Thomas, G., Jour, of the Australian Institute of Metals, v. 8, p.

8 0 , 1963

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Major Impurities a/o


99.999 +


Sb Pb Sn

,0001 .0001 .0001



,0001 .0002

Te Se

,0002 .0001




99.999 +






Parts per Million

Cu Pb Mg



Mn Si Fe


^Reported by suppliers




Analysis, w/o Zn

Twin/Grain Energy

+ t




2 .6 0


6 .8 5


1 0 .8 8

1 2 .8 8

0 .0 7 6 + .014

0.042 + .012

0.041 + .009

0.035 + .001

0.043 + .015




0.039 +


Grain Boundary Energy ergs/cm

2 (Taylor, 1 9 5 8 )









Twin Boundary

Energy, ergs/cm





2 3 ,6



2 1 .3


w/o Zn





2 .6 0


6 .8 5

9 .6 0

1 0 .8 8

1 2 .8 8

1 5 .0

2 0 .0

2 5 .0

3 0 .0


0 .6 2




.8 9



.8 5







Ln Intensity





(Howie & Swann,






0 .6 0


0 .1 2

0 .1 6













3 6 .6

















Fig. 1. Ball model of (a) splitting

of total dislocation into partials; (b),

(c) resulting fault.



Pages 75 through 77 do not exist.

(error made in typing page numbers).


Pure Copper

x = 0.076







1.03 w/o zinc

x ■ 0.042




0 L -







Fig. 2. Distribution of values of twin/grain boundary energy for pure copper and 1.03 w/o zinc in brass.

2.60 w/o zinc

x = 0.041


30 -0.15


0 i



15 r

5.30 w/o zinc

x - 0. 035




0 0.15



Fig. 3. Distribution of values of twin/grain boundary energy for 2.60 w/o zinc and 5.30 w/o zinc in brass.



6.85 w/o zinc

x = 0.043







12.88 w/o zinc

x = 0.039






Fig. 4. Distribution of values of twin/grain boundary energy for 6.85 w/o zinc and 12.88 w/o zinc in brass.


& 0.06






12.88 w/o zinc






0.06 r

H w g 0.04



M 0.02


50 80






Fig. 5. Variation of mean and range of confidence with number of

values of twin/grain boundary energies.



53 0.06

g 0.04

a 0.02




CM s v 560





CQ o 540


T a y l o r , . > 5 8











Twin and

Grain boundary energies for

Cu-Zn alloys.


Pure Copper

1.031 Zn o



2.601 Zn

5.301 Zn


7. fill)

pole figures of high purity copper and brass rolled

98.31. 1 zinc in brass as indicated. ■ fllO) <112> A (53]} <001>


6.85% Zn

9.601 Zn


10.88% Zn

12.88% Zn

PI r

. 8.

till) pole figures of high purity brass rolled 98.3% .

% zinc in brass as indicated. ■ fillO} <112> A {531) <001 >

• s o .

400 ijoo' zoo ioo

96.8-3.2 Cu-Zn

93.8-6.2 Cu-Zn

89.7-10.3 Cu-Zn

Fir,. 9.

fill} pole figures of brasses rolled 96%. Weight percent zinc in brasses as indicated.

(After Merlini and Beck, 1955.)











Fig. 10. Variation of texture intensity ratio with weight

percent zinc in brass.

XPHA = 0




Figure 11. Partial {lll\ pole figure scan of 99.99% copper rolled




Figure 12. Partial (lll^ pole figure scan of

99.999^ copper rolled 98.30







□ Howie and Swann (1961)

O Values obtained in this




0 5




20 25

Fig. 13. Apparent stacking fault energies for Cu-Zn alleys.

Corrected results of Howie and Swann (1961) are plotted for com­ parison.


40 50 60


E N E R G Y ,


70 80







30 40

50 60


70 80

Fig. 14. Variation of texture intensity ratio with stacking

fault energy.





The basis for chemical analysis by x-ray fluores­

cence is that each element will emit its characteristic

lines when, bombarded by high energy x-rays* Every element

has a characteristic line spectrum. The most intense lines

of this spectrum are the K-alpha and the K-beta.

The analysis is carried out with the aid of an x-ray

spectrometer by diffracting the radiation from crystallo-

graphic planes of known d-spacing in a single crystal.

This single crystal> most commonly a LiP crystal, enables

the characteristic wavelengths emitted by the specimen

to be recorded one at a time.

The analysis may be qualitative or quantitative.

In the former method, all that is required is a scan of

all possible angles from

6 = 0 to 9 - $0°. The. angles at

which intensity peaks occur may then be related to elements

which produced the peaks by referring to published tables

such as "X-ray Wavelengths for Spectrometer", General

Electric Co.,

1 9 6 1 . In the latter method, the intensities

of the peaks are compared with intensities of lines from a

suitable standard.

The steps involved in analyzing alpha-brasses were

as follows:

1 .

Preparation of copper and zinc standard



Determination of optimum setting of x-ray



Determination of fluorescent intensity versus

percent copper and percent zinc for the

standard solutions.

4. Preparation of solutions from ingots.


Determination of zinc content of ingots by

comparing intensities with the standard


This zinc analysis was made using the G-E XRD-5 X-Ray


The brass standards were prepared by mixing the

desired proportions of copper and zinc solutions that were

prepared from the pure metals to a concentration.of

0 .0 1

gm/cc. .For example, a 94-6 Ou-Zn standard solution was

prepared by measuring 4? cc of copper solution and adding

3 cc of zinc solution.

The samples were placed in a plastic container with

a Mylar window to hold the solutions while they were being


It was found that it was necessary to use a

constant volume such that a flat surface was produced at

the Mylar window to obtain reproducible results.

The x-ray variables in this method are (a) KFP,

(b) milliamperes, (c) target, (d) analyzing crystal, (e)

Counter type, (f) 29 values, (g) counter voltage, and

(h) pulse height selector.

The KVP was the maximum voltage recommended for the

target used, 50 KVP. The milliamperage was determined as


that current which would enable both copper and zinc

Intensity peaks to remain within the range of the chart

recorder scale regardless of their concentration. The

target used was platinum since it was the only one available

but proved satisfactory since no platinum peaks appeared

near the Cu K-alpha peaks. The analyzing crystal, LIP, was

satisfactory because the K-alpha peaks of copper and zinc

appeared sharp and intense. The counter voltage was deter­

mined by obtaining a counter plateau curve, using the same

material as that to be analyzed. The counter voltage was

then chosen at about midpoint of the plateau. The 2©

values were determined by obtaining

2© scans over the range

of K-alpha peaks found in.tables. These are:








Zn K-beta

Cu K-beta

Zn K-alpha

Cu K-alpha

Only the K-alpha peaks were used because these were of the

higher intensity. The pulse-height selector filters out

reflections which occur near the desired peaks. This was

found unnecessary in this case but a voltage window was

found desirable to produce sharper peaks.

Below is a listing of all x-ray variables:






Analyzing Crystal

Beam Slit

Counter Voltage

- 50

- 5

- Ft

- Scintillation

- LiF

- 0.010° Seller

- 1.3 DC KV

Using the K-alpha peaks for Cu and Zn, the ratio of

Intensities of the Zn K-alpha to the Cu K-alpha was obtained.

This ratio eliminates an internal standard and corrections

for day-to-day fluctuations, as well as corrections for any

secondary emissions.

Intensities were determined by setting

the x-ray goniometer to the K-alpha peak and obtaining the

counts per second by use of a scaler. The period of counting

was 1000 seconds. Values obtained, in counts per second,

are plotted in Figure A. The values of the ratios vary by


0 .1 5 w/o zinc from the curve determined by the method

of least squares. Therefore, the overall accuracy should

be within this value.

To analyze the ingots of brass, filings from each

end of the ingot were obtained and dissolved in a

50$ nitric

acid solution after which they were diluted to the same

concentration as the standards. The analysis then consisted

of comparing the Zn K-alpha to Cu K-alpha intensity ratio

against the standard curve.









Fig. A. Variation of intensity ratio of zinc K-alpha to copper

K-alpha with increasing zinc content.





When a pinhole x-ray pattern of a polycrystalline

metal is obtained, the Debye rings which appear are the

result of aggregations of spots formed by reflections of

{hkl} planes. Each spot corresponds to a particle of

powder or a grain which has its {hkl^ planes oriented so

as to make the correct Bragg reflection. In a metal whose

grains are oriented randomly, the population of spots around

the circumference of the Debye ring is equally distributed

resulting in a ring of equal intensity around its circum­

ference, When the metal has its grains preferentially

oriented, however, the Debye rings are of nonuniform

intensity or even discontinuous around the circumference.

The low intensity or missing portions of the rings are due

to few or no grains being oriented at those positions for

Bragg reflection.

The degree and type of preferred

orientation or texture, then, may be determined by analysis

of the positions and intensities of the Debye arcs.

A diffractometer counter tube counts the diffracted

x-ray quanta from a portion of a Debye ring.

In order to

use this device for determination of preferred orientation

of a specimen from a rolled sheet, the specimen must be

rotated about an axis perpendicular to its plane (beta

angle) so that the entire Debye ring (or arc, in the case

of textured specimens) passes through the counter window.


In addition to this rotation, it is necessary to rotate the specimen about an axis in the plane of the specimen

(alpha angle), since the texture does not have rotational symmetry about the rolling direction. The latter rotation can be explained further by considering the planes contri­ buting to a given \hkl$ diffraction.

When the specimen is placed in the pole-figure goniometer for transmission at alpha equals 0°, the speci­ men plane bisects the angle between the incident and trans­ mitted beam as shown in the sketch on this page. The intensity measured at this alpha angle is due to diffraction from {hklf, planes which lie normal to the plane of the vertical specimen holder ring. The pole of the specimen surface lies at the center of the polar net and the poles of the {hkl^ planes, therefore, lie on the periphery of the net. The polar net corresponds to the face of the specimen as viewed from the counter.

Strip specimen pole of (hkl)

X-ray beam



If the specimen is rotated about the diffractometer axis from its zero position by 5 ° alpha, the intensity

measured is the result of diffraction from ihklj planes


5 ° from the normal to the plane of the vertical

ring. This corresponds to a rotation of the £hkl$ poles

also of 5° along a meridian. The pole of this rotated

plane, then, lies at

5 ° from the periphery towards the center

of the net.

When a scan, of a textured specimen is made through

several alpha angles as described above, the data can be used to plot a stereographic view of the distribution of

{hkll planes for the specimen.

Prom these pole-figure

plots, information such as type and degree of preferred

orientation, deviation from ideal textures, comparison of

mode of deformation between metals and alloys, texture

transitions, etc. can be obtained.

Since the intensities obtained in pole-figure scans

are dependent upon such factors as machine characteristics

and specific techniques used, it is desirable and common

practice to use intensities based on a sample of the same

material of random orientation. This tends to afford

reproducible results independent of different machines

and techniques,

A detailed discussion of the various specific pro­

cedures involved in obtaining a pole figure is offered below

and arranged according to the following general topics:


A c Preparation of Random Sample

B. Determination and Setting of Optimum Machine

Operating Conditions

C, Obtaining Data from Random Sample

D„ Preparation of Textured Sample

E. Obtaining Data from Textured Sample

F„ Analysis and Plotting of Data on Polar Net

The procedures described here are based on the

preparation of pole figures using a General Electric XRD-5

X-ray Diffraction Unit equipped with a General Electric

Automatic Integrating Pole-Figure Goniometer.


A. Preparation of Random Sample

There are several ways of preparing a randomly

oriented sample. Among these are spraying of the powder

mixed with Glyptal varnish (Geisler, 1953) or using gravity

settling of powder through molten paraffin followed by

firing of the paraffin-impregnated powder (Newkirk and

Bruce, 1957)« The following method adopted in this labora­ tory has also proved to be satisfactory.

A.I. Machine a piston-cyUnder die with the inside

diameter of the cylinder equal to the diameter of the speci­ men holder on the pole-figure goniometer. Actually, a some­

what larger diameter would be more satisfactory to eliminate

edge etch effects on the finished specimen.



2 „ With the piston in the cylinder, place enough

powder to obtain a thick compact (about

1 / 8 in. or less of

loose powder). A thinner compact will result in a textured

sample. Using a heavy-duty laboratory press, prepare the

compact. Sinter the compact, then grind the specimen to

the standard thickness. The specimens should be reduced

as far as possible mechanically and then etched to the

desired thickness. The thickness should be enough to pro­

vide essentially complete absorption of the primary beam

for the reflection method but not much thicker than about

0.010 in.

It is possible to use the same specimen for

both transmission and reflection if the thickness is less

than about 2/y<u., w h e r e i s the linear absorption coeffi­ cient (ASTM Std. E81-54T, Pt. 3, 196l).

B. Determination and Setting of Optimum Machine Operating


Target -- The choice here is dependent upon the

absorption of the test material.

KVT-MA -- This is chosen to obtain maximum intensity

without exceeding the x-ray tube limitations.

Beam Slit — Slits are chosen as a compromise

between intensity and resolution. The wider the slit, the

higher the Intensity at the expense of resolution. The

beam width at the beam slit should be sufficiently narrow

to prevent the beaSm from striking the specimen holder at

any alpha anglea This slit should also be wide enough to

provide adequate intensity for pole-figure traces.

It may

be necessary to mask off the beam at this slit to prevent

the beam from striking the top and bottom of the specimen

holder as it oscillates.

Pole-Figure Goniometer — This device automatically

provides the necessary rotations as explained previously.

Beta rotation — is rotation of the specimen about

an axis normal to the specimen surface in angular increments


16 minutes of arc. This rotation is at the rate of one

revolution in 45 minutes, and is clockwise as viewed from

the counter.

Alpha rotation — consists of rotation of the speci-

men holder in discrete

5 steps about the goniometer's

vertical theta axis. The rotation is clockwise as viewed

from above.



Determine the alpha ranges to be used in

transmission and reflection.

In transmission, alpha equals

zero when the specimen holder pointer is at zero. The alpha

range is limited by beam interference from the specimen

holder. The available ranges depend also on the 29 angle

being used. For greater range in the overlap region, it is

advisable to use a higher order plane in reflection. For

104 aluminum, using (ill) for transmission and ( 3 3 3 ) in reflec­ tion, the following ranges were possible: transmission

0 ° to 5 5 ° alpha, reflection -- 9 0 ° to 5 5 ° alpha„


Determine the number of quadrants necessary

to investigate in order to obtain the complete pole figure.

Most types of deformation produce a

1 8 0 ° symmetry and in

many cases a

9 0 ° symmetry; that is, one quadrant is suffi­

cient to describe the whole pole figure, in the latter case.

In some cases, however, it may be desirable to use a full

360° of beta rotation at each alpha angle. When this is

determined, set the beta range on the goniometer controller.

3. Set beta at 0° and insert specimen into the

specimen holder with the rolling direction vertical. This

results in having the rolling direction at beta =

9 0 ° on

the polar net since the portion of the Debye arc at the

counter lies at a right angle to the rolling direction.

4. Set alpha at

0 ° for transmission or at 90°

for reflection.

5., Press ready button until green light turns on, on the controller panel, then press once more.

In the case


9 0 ° beta increments are used, it is unnecessary to

use the ready button.

6 .

Begin pole-figure trace by simultaneously starting recorder chart and pole-figure goniometer.


Detector Slit -- Since the alpha rotation alters

the geometry of the focusing circle, the detector slit does

not increase the resolution by very much at alpha angles

greater than zero. The detector slit also cuts down the

width of the peak entering the counter. For the pole-figure

preparation procedure described here this is not an important

factor. However, when calculating intensities for alpha

angles other than at zero degrees, as suggested by Decker,

Asp, and Harker (1948), it is necessary to use integrated

intensities and, therefore, the whole width of the 29 peak

must enter the counter.

being employed. In incident beam since the highest ratio of V l to background ls Siven in this arrangement

(Newkirk and Bruce, 1957).

Chart Speed -- The chart speed to use is almost an

arbitrary choice, but the best compromise.between using a

large amount of paper and having too much peak convergence

seems to be at

12 in./hr.

Chart Scale -- The log scale is most commonly used.

The use of this scale necessitates a log ruler to be pre­

pared by reading several intensity counts along the scale

after calibrating the scale to read

1 0 ,0 0 0 counts at full

scale. Notet the log scale on the chart is not a base

r,e" or base "10" but is an "electronic" log scale. The

linear scale can also be used when the highest peak


intensity lies within the range of the scale used. This

scale is sometimes more convenient than the log scale because

the intensities are easy to read, the scale is simple to

calibrate, and the low intensity insignificant peaks are

not so prominent,

X-Ray Unit Stabilization — When the unit (power

and tube) is cold, a markedly higher intensity will be

recorded. As the unit warms up, the intensity drops off

in an exponential manner and levels off after about 40

minutes of operation.

The line voltage and tube, then,

should be run approximately 45 minutes before performing

any operation which involves Intensity measurements.


a standard intensity sample is available, it is more con­

venient and accurate to use this than to wait for a specific


Specific Conditions -- The following conditions

were used for obtaining a pole figure of aluminum:




COUNTER Proportional




Ni (in filter slot)


0 ° to 55°



BETA RANGE 0° to 360° 2 0 SPEED 12ll/hr.




° (




The same conditions were used as above except

for the alpha range which in this case was

9 0 ° to

5 5 ° p a change in the filter placement to a position

in the diffracted beam, and a different 2 9, in this


1 6 2 .5 ° for the (3 3 3 ) reflection.

C. Obtaining Data from Random Sample

If this sample is to be used subsequently as a

standard intensity sample, it is desirable to allow a

longer warm-up time for line and tube than prescribed



Randomness -- The randomness may be checked

either by a pole-figure trace or by a pinhole x-ray photo­ graph. The former is the more exact method.

<3.2. Absorption factor, /xt — The procedure to

be used is as follows: a. Using the identical

20 and slit and filter

arrangements to be used for pole-figure scans, set a diffrac­

tion specimen, preferably of the same material, on the

specimen holder. Determine the counts/sec. on the scaler.

This figure is 1^.


Set the sample in front of the counter

window and obtain the counts/sec. This figure is I.


From several readings, or longer counting c. time, determine the mean

yut for the sample by the relation:

I = I0 e" ^

Note: The t will not be the same for different 20 values due probably to phenomena such as secondary emission.

C.3. Background Intensities — The purpose of

obtaining these background intensity measurements is

ultimately to plot and report intensities above background

and thereby produce reproducible pole figures which are

independent of machine and operating conditions.

For the chosen 20 angles in transmission and reflection, the background is determined for all alpha angles to be used. When no overlapping peaks are present at the chosen 20, these backgrounds may be determined by either of the following methods: a. Obtain a slow rate 20 scan to include portions on each side of the peak and note the trend of the background intensities. Use the interpolated value as background intensity.





a o

20 -4

b. alpha =

0 , calculate the rest of the alpha backgrounds by

the intensity correction, I

0/Iy<< , which can be obtained

from the data in step P below.


Using the value of background for only

In the case where overlapping peaks occur in the

near vicinity of the 29 peaks, Acucena and VanKuren (1958)

outline a background intensity determination method.

If the background intensity from the random sample

is above the lowest trace given by the textured sample,

the random sample is not a true random representation.

C.4. Lattice Parameter -- A comparatively large

difference in lattice parameter or, more specifically, a

large difference in line position of a textured sample as

compared with the random sample, will cause the intensity

of the textured sample to be too low at the 29 angle used

for the random sample.

It is convenient, therefore, to be

able to determine beforehand if a pole figure of a certain

alloy of the random sample material may be prepared using

the same random sample. A knowledge of the lattice para­ meter of the random sample and the alloy in question accom­ plishes this purpose.

Even when the lattice parameter does not change

significantly, however, slight apparent peak shifts can

occur with different alpha angles. The difference in

intensity on the pole-figure scan can be as high as a

factor of two. If this "peak shift" is not equal for

different conditions of specimen treatment or alloying,

then this can affect the interpretation of a series of

pole figures.


D. Preparation of Textured Sample

The preparation of the textured sample depends, of

course, on the starting and ultimate material conditions

desired. An example will be outlined herein, in which a

cast aluminum ingot was reduced, rolled, and prepared as a

pole-figure specimen.

The specimen from the ingot was first alternately

compressed to a predetermined reduction and annealed until

it could be accomodated by the available rolling mill.

The reductions were determined by the relation:

% R = 1 - ( F / t ^ V n ) , where

$ R = percent reduction

P = final thickness

I = initial thickness

n = number of passes.

The reduction per pass, or compression operation, was deter­

mined by the work hardening of the material used, as well as

by the capacity of the press being used.

The specimen was checked, at this stage, for texture

by a pinhole pattern or pole-figure trace. Since the speci­ men was to be rolled less than

90 %, it was required that it

Ill be all or nearly texture free.

If a specimen Is to be

rolled greater than

90$, the compression texture will prob­

ably disappear. The specimen was then rolled to the desired

reduction and a section was punched out, using the powder

compact die. The punched-out specimen was then ground and

etched to the absorption factor of the random sample»

Alternate methods of reducing the specimen are by

chemical and electrolytic techniques. In the latter, how­ ever, reaction products may form on the surface and conse­

quently may cause spurious peaks to appear in the pole-figure


It is probably possible to etch this surface layer

off. In both methods there is danger of losing a consti­

tuent, in the case of alloys, due to preferential action

on the specimen.

E. Obtaining Data Prom Textured Sample

E.l. Lattice Parameter — The lattice parameter

is determined for the purpose previously mentioned in C.4.

E.2. Absorption Factor, /-tfc — The absorption

factor for the textured specimen is determined in a similar

manner to the. random sample. In addition, it may be neces­

sary to obtain I readings at short beta intervals due to

absorption differences. Newkirk and Bruce (1957) report

that the apparent yOct of a textured specimen (of Pe) was

found to vary as much as 45$ with beta orientation.


high-purity Al, 7178-T6, and Cu, the

y c c t did not vary

112 significantly with beta orientation.

In the case where

the t varies with beta, a weighted average of the

/ J t

values should be used.

E„3o Background Intensities — These are determined

in a manner similar to that described for the random sample.

E.4. Pole-Figure Traces — The following steps

may be taken to obtain the pole-figure traces: a.

Set alpha at zero and allow beta to rotate

through 9 0 °, 1 8 0 °, or 3 6 0 ° on a certain 26.


Correlate the motion of the recorder

chart with the beta rotation, and mark the prominent direc­

tions in the specimen (such as the rolling direction) on

the chart to be used as the origin for measuring angles.

F. Analysis and Plotting of Data on Polar Net


1 . Random Sample Chart Preparation — Details of

the procedures for transmission and reflection are given


a. Transmission;

A convenient form of tabulation of data

is the one on the following page.



Alpha Counts Random Random I Textured IX


BG Above BG BG


6 I :

Use alpha range as determined in step B„

Cole II, III, IV ; The intensity at each alpha

angle is determined by counting continuously through the

beta range used. This is accomplished by setting the scaler

on "Manual" and using the "Stop" lever when the indexing

occurs. Using the time recorded on the scaler, the average

ops value is determined. Another method is to draw an

isointensity line on the recorder chart which best averages

the intensity trace.

Col. V : These are determined as outlined in step

C.3. above„

Col. VI : Subtract value in Col. V from value in

Col. IV.

Col. VII : These are determined as outlined in

0 .3 . above.

Col. VIII : Add value in Col. VI to value in Col.


Col. IX : Twice the value in Col. VI plus value in

Col. VII, Other multiplicities of random sample intensity

are used which are estimated to intersect peaks on the pole

figure trace. Each multiplicity should be marked with a

different color which will aid in identification when

plotting the pole figure.

I at IX = Random I Above BG plus Textured BG.

I at

2X = 2 (Random I Above BG) plus Textured


Mote that I at

2X is not equal to 2 (I at IX).

b. Reflection:

A convenient form of tabulation of data

in this case is given below.


Alpha Counts Seconds CPS Random Random I Corr. for

BG Above BG Overlap


Text. IX

The columns in reflection are similar to those in

transmission with the exception of Col. VII. The use of

equicolor multiplicities for different alpha angles in

transmission, as well as in reflection, provides the neces­

sary correction for increase in absorption and diffracting

volume with alpha angle. In effect, in the case of trans­

mission, all alpha angle intensities are increased to the

value of the intensity at alpha equals zero.


in reflection all alpha angle intensities are decreased to

the value of the intensity at alpha equals 90°.

It is then

desirable to conciliate the intensities in reflection to

those in transmission.


To accomplish this conciliation two methods are

suggested: (a) the use of a correction factor on reflected region intensities I

0°/ I <?c = 90°; or (b) preparation

of samples such that these intensities will be equal. In

the former method, even after applying the correction factor,

the transmission and reflection intensities may not coincide.

In the latter method it is necessary to thin the specimen

until the transmitted intensity equals the reflected inten­ sity, This may not be possible for all metals,

F,2, Plotting the Pole Figure — a. Make a calibrated horizontal line may be measured in beta degrees, b.

The beta angles at which each trace

crosses an intensity level are measured with this rule,

and a point at the corresponding alpha and beta coordinates

is plotted on a polar stereographic chart in a color

characteristic of the intensity level. The polar chart has

beta degree graduations on its circumference and increasing

alpha degree graduations in receding concentric circles.

It is advisable to plot all arcs of one peak for all the

alpha angles and to join these arcs into one contour before

plotting other peaks to prevent confusion in drawing the

contours. Illustrations of these measurements, as well as the final contour lines are shown on pages 148-151

"Modern Research Techniques", A.S.M., 1953»

c. Sequence of Plotting: In transmission,

with alpha set at zero, the beta rotation is clockwise

as viewed from the counter. This causes the Debye arcs

to pass across the counter window in a counterclockwise

sequence from beta equals zero, as indicated below.

1 1 6 beta beta

Debye arc crossing

For l80° symmetry, the pole figure is plotted in the

following sequence: beta = 0


0 alpha = 0 10

alpha = 0

alpha = 0 __

- 180


In reflection, for l80° symmetry, the pole figure

is plotted in the sequence shown below:


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