Magnitude of the zener relaxation effect—III anisotropy of the relaxation strength in Ag-Zn and Li-Mg solid solutions

Magnitude of the zener relaxation effect—III anisotropy of the relaxation strength in Ag-Zn and Li-Mg solid solutions

MAGNITUDE ANISOTROPY OF THE OF THE RELAXATION ZENER RELAXATION STRENGTH IN A&2x1 EFFECT-III AND Li-Mg SOLID SOLUTIONS* and A. S. NOWICK ...

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MAGNITUDE ANISOTROPY

OF THE

OF THE

RELAXATION

ZENER

RELAXATION

STRENGTH

IN A&2x1

EFFECT-III

AND

Li-Mg

SOLID

SOLUTIONS*

and A. S. NOWICK

D. P. SERAPHIM

The anisotropy of the Zener relaxation effect is determined in face-centered cubic Ag-Zn and bodycentered cubic Li-Mg solid solutions from measurements made on oscillating single crystal rods both in flexure and in torsion, The effect is found to have a high anisotropy, such that the relaxation strength A0 is greater for (111) than for (100) oriented crystals. This direction of the anisotropy applies to both the f.c.c. and the b.c.o. solid solutions. The relaxation strength for a given orientation varies very nearly as the square of the solute concentration for the Ag-Zn alloy up to 30 at.% Zn. Finally, the existence of dilatational relaxation is demonstrated for the Li-Mg alloy. Neither the pair-reorientation theory nor the directional ordering theory, which are based solely on nearest neighbor interactions, is capable of explaining the observed anisotropies. In fact, it is shown that an important role must be assigned to relaxations involving next-nearest neighbor atom pairs in order to explain the observed effects. 1MPORTANCE DE L’EFFET DE RELAXATION DE ZENER-III. ANISOTROPIE DE LA FORCE DE RELAXATION DANS LES SOLUTIONS SOLIDES Ag-Zn et Li-Mg Les auteurs ont determine l’anisotropie de l’effet de relaxation de Zener dans des solutions solides cubiques a faces centrecs Ag-Zn et cubiques centritesLi-Mg. Les mesures ont ete effect&es sur des fils monocristallins oscillants en flexion et en torsion. L’effet de relaxation presente une anisotropie importante: la force de relaxation A, est ainsi plus grande pour les cristaux orient& suivant (111) que pour ceux orient& suivant (100). Cette direction d’anisotropie est valabIe aussi bien pour les solutions solides cubiques faces cent~rcies que cubiques centrees. La force de relaxation pour une orientation determinee varie quasiment comme le car& de la concentration en solut&pour Ies alliages Ag-Zn contenant jusque 30 ch at. de zinc. Enfin, l’existence d’une relaxation de dilatation est prouv&epour les alliages Li-Mg. Pour expliquer les orientations observees, les theories de l’ordre directionnel ou de la reorientation par pairc sont inadaptees. Ces theories en effet sont baseas uniquement sur les interactions entre plus proches voisins. En fait, les auteurs montrent que pour expliqer les effets observes, uu role important doit btre attribue aux relaxations des paires atomiques constituees du proche et du plus prochc voisin. GRGSE DES ZENER-RELAXATIONSEFFEKTES-IIT ANISOTROPIE DER RELAXATIONSSTARKE IN FESTEN VON Ag-Zn UND Li-Mg

LGSUNGEN

Die ilnisotropie des Zener-Relaxationseffektes von festen Losungen von kubisch-flachenzentriertem Ag-Zn und kub~sch-ra~zentrier~m Li-Mg wird aus Messungen an schwingendan ~i~r~talIs~~~~ in Biegung und Torsion be&runt. Der Effekt ist sehr anisotrop, und zwar ist die Relaxation~t~rke A0 fiir Kristalle in (ill)-Orientie~ng grb;lJerals fur (100); das gilt fur die flachenzentrierten wie fiir die raumzentrierten Legierungen. Fiir gegebene Orientierung variiert die Relaxationsstiirke bei der Ag-ZnLegierung ziemlich genau wie das Quadrat der Zusatzkonzentration bis au 30 Atom% Zn. SchfieBlich wird die Existenz einer Dilatationsrelaxation fur die Li-Mg-Legierung nachgewiesen. Weder die Paar-Oriontierungstheorie noch die Theorie der Ordnung mit Vorzugsrichtung, die nur auf der Wechselwirkung niichster Nachbarn basieren, kann die beobachtete Anisotropie erkliiren. Es wird sogar geziegt, da13 man Relaxationsprozessen, die Paare iibernachster Nachbarn betreffen, eine bedeutende Rolle bei der Erkliirung der beobachteten Effekte zuschreiben muB. 1. INTRODUCTION

The program

of investigation

relaxation strength, A,, phenomenon on various

of the dependence

of

for the Zener relaxation parameters was begun in

papers I and II of this series,(ly2) for the purpose

of

testing the existing theories of the effect. Paper I dealt with the variation of relaxation strength with alloy system, while in paper II the dependence of AM on temperature was studied for the u-Ag-Zn solid solutions. One of the remaining objectives is to study the anisotropy

of the relaxation

strength,

i.e. the

* Received February 15, 196Oj revised June 27, 1960. 7 Formerly at Yale Universrty; present address: IBM Research Center, Yorktown Heights, New York. ACTA METALLURGICA, 1

VOL. 9, FEBRUARY

1961

dependence of A, on work(a) on polycrys~~ine

tive evidence

crystal Ag-Zn

for such anisotropy,

orientation. Early wires gives qualitaby showing

that

wires of different crystal textures give substantia~y different relaxation strengths. Such results show the limitations

of working

with polycrystalline

samples,

namely, that precise quantitative comparisons cannot be made between different samples without some knowledge of the mean grain orientations. For this same reason, the concentration dependence of the relaxation strength has only been determined roughly in previous work.@p4) CIearly the anisotropy of the relaxation strength is best studied by working with single 85

crystals

of

known

orientations.

The

only

ACTA

86

previous

experimental

METALLURC+ICA,

work on single crystals is that

of Artmanc5) who dealt with partially ordered j3-brass (b.c.c.) crystals and reported essentially no dependence of relaxation strength on crystal orientation. Unfo~una~ly the analysis of Artman’s data is complicated by the strong change of long-range temperature

and by the precipitation

order with

of the alpha-

phase in some specimens. The present paper has as its objective the study of the variation of relaxation strength with orientation for single crystal

rods of solid solutions

f.c.c. and b.c.c. types.

VOL.

9,

1961

2. FORMAL DEPENDENCE

THEORY OF ORIENTATION OF RELAXATION STRENGTH

By virtue of the symmetry of cubic crystals, their elastic properties are completely determined by the three elastic coe~~ients(s)

srl, srs and s44, For present

purposes it is convenient of elastic coefficients.

to introduce

notation two compressibility

of both the

s = sa4

The cx-Ag-Zn alloys were again

large effect,

and because

menon for this system. advisable

For the b.c.c. alloy, it seemed

to select a system not complicated

onset of long-range

by the

order or other phase transforma-

tions at the lower temperatures.

This requirement

rules out most of the @phase

alloys of the CuZn type.

Interest in the b.c.c. Li-Mg information that an internal

alloys was aroused by friction peak had been

found*

in an investigation

of damping

capacity

tion suggested,

this peak.

and subsequent

.

(1)

1

Consider now the case of a crystal in the form of a cylindrical to obtain

rod which is subjected a value for Young’s

to flexure or tension modulus, E, or to

torsion to obtain the rigidity modulus,

G. The formal

theory of elasticity gives the dependence of these moduli on the orientation of the cylinder axis and on the three elastic coefficients, in terms of the well known equations.ts)

for

This descrip-

investigation

/

8i.J

sn = (sn _t- 28,s)

the purpose of eliminating vibrations in some engiapplications. The large grain-boundary neering damping, which produced a rapidly increasing internal friction at temperatures higher than that of the peak of interest, almost obscured

1

s’ = 2(&r, -

there exists

considerable information concerning the kinetics(3*6*7) and the magnitude(2) of the Zener relaxation pheno-

a different set a widely used

shear coelEcients, a and s’, and a coefficient, s”, are defined as follows:

chosen for thef.c.c. solid solutions because they showed a particularly

Following

G-l = s + 2(s’ -

s) I’

(3)

where r is given by

showed,

that the peak itself was due to a Zener relaxation.

r = (ar2as2 + a12o132+ az2as2)

(4)

Thus, Li-Mg was chosen as the b.o.c, alloy for investigation

of the orientation

dependence

of relaxation

and the X’S are the direction

cosines of the axis of the

strength. It will be shown in this paper that the measurement

specimen relative to the principal axes of the crystal. The quantity E-l is thus linear in I’ with negative

of relaxation

slope (a’ -

provides

strength

as a function

of orientation

a more severe test of current theories of the

Zener relaxation measurements measurements

phenomenon

than any of the other Such to this point.

reported up also yield information

~ntration dependence given orientation.

of the relaxation

Regardless of mechanism, theory of the relaxation

on

the

con-

strength for a

however, there is a formal

strength and its dependence

a) and intercept

(s’ + s”)/3,

while G-1 is

also linear in I’ with twice the slope (but opposite

in

sign) and intercept s. It is now necessary to obtain expressions for the relaxation strengths, AE and A,, for oubic solid solutions

in terms

coefhcients relaxation

of the relaxed and unrelaxed s, s’ and sn. From equation (I-11)7, the strength for Young’s modulus may be

written:

on crystal orientation which is as firmly grounded as the theory of the dependence of the elastic constants

h

=

E

f-J’/’- E,-’ _ E,-1

dE-1 E,-1

on crystal orientation. This formal theory has been presented previously by Zener’s), but it is repeated here (see Section 2) in a slightly different form which is very useful in checking for the self consistency of

re~~ut~on for the reciprocal Young’s modulus. Since E,-l must obey equation (2) with the relaxed

the experimental

s-coefficients

results.

* The authors me indebted to J. 33. Clark of the Dow Chemical Company for this information.

where

6E-r =z ET-l -

E;-1

will

(s,., s,‘, s,“), while E;-l

be

termed

must obey

the

the

t The notation (I-n) and (II-n) will be used herein to refer to equation (12)in paper I and II, respectively.

SERAPHIM same equation

AND XOWICK:

with

ZENER

the unrelaxed

RELAXATION

coefficients

(su,

STRENGTH

IN Ag-Zn

was used in calculations In all, 45 Ag-Zn

!!f’_ ~+3 “’ -

(6s’ - 6s)

r

(6)

where 6s = s, -

s,

(7)

and similarly for 6s’ and 6s”. With similar treatment, the relaxation strength for the shear modulus is A0 = 6G-r/G,-i where 6G-i = G,.-1 for the reciprocal of relaxed

0,-l

(8)

is defined as the relaxation

rigidity modulus.

and unrelaxed

The substitution

expressions

from equation

(3) then yields:

equations

-

6s) I’.

(6) and (9) for the relaxations

for the relaxation

criteria for selection

Single crystals of Li-Mg technique,

The procedure

is described

paper.@)

These

G-f = $8+ 2(dS’- 8s)r a,, + w,’ - su) r

G,-1

Clearly, since the relaxation of two linear functions vary

linearly

equations

relaxation orientations,

crystals.

were

The composition

determined,

with

l?.

It

to be close to that of

must

occur

at

shown

from # 0

the

extreme

(100) and (111).

back reflection

of Ag-Zn

Bridgman

Laue data.

times three or four,

alloy were grown by the

technique

cosines of

to the three principal

axes were measured from stereographic

cubic

plots of X-ray

At least two, and some-

pictures

were taken

along

the

In general the sums of the squares of the three cosines measured

independently

.002, and the several

Single crystal in a free-free

specimens mode,

was equal

stereographs

for each

within l/Z’.

in graphite

molds

enclosed in glass and then sealed off. In each case, a length of 4 cm was removed from each end of the crystal leaving 9 to 11 cm of the center section which

were oscillated

by a magnetic

described

elsewhere.

be obtained

method

The internal

friction

can then

in the usual manner(s) from the width of

the resonance

in forced vibrations.

from

strength

the maximum

It would appear

can be obtained value

most

of the internal

friction as a function of temperature, see equation (I-8). A second simple expression for t,he relaxation strength comes from the difference between unrelaxed and relaxed moduli-see equation (I-l l)-remembering that m2 is proportional to the dynamic modulus, where w is the circular frequency

at resonance.

were relatively homogeneous in composition. A typical analysis* at 4 points on the center section of a near 25 at.% Zn is: 25.1, 24.6, 24.7, and 24.7 at. V ,0 Zn, at locations 0, 0.4, 0.6 and 1.0 of the length, respectively. The crystals, originally of 0.325 cm diameter, were

in flexure

drive

similar to that originally employed by Ke(lO). The magnetic drive method employed for torsion will be

that the relaxation

3. PREPARATION AND ANALYSIS OF SPECIMENS Single crystals

axis relative

4. PROCEDURE

of interest. Thus AE and A, vary with r, i.e. extreme values of the

strength

by the :Dow Chemical

(11)

*

it does not in general is readily

polished

of these crystals was not

but was assumed

the crystal

strength is the quotient

in r,

chemically

are

directly

conventional

crystals

steel molds.

in more detail in another

specimen resulted in the same orientation

(10) and (11) that in general, dA/dI’

in the range monotonically

alloy also were grown by in low-carbon

and then measured in the same manner as the Ag-Zn

direction

_

(b) length

(d) absence of surface grains.

the Bridgman

to 1 f G

were (a) orientat’ion

(c) surface smoothness

(9)

strengths are

1.5%,

16.6 f 0.3%, 10.6 3 0.4, and 3.70& Several crystals in each range were chosen for measurements. The

length.

A

viz. 30.25 $I 1.5%, 26.05 f

Company, which was 57 at.O/ Mg. For both types of crystals, the direction

linear with I’, but 6G-l has a slope of opposite sign to dE-1 and of twice the magnitude. The full expression

The

crystals were grown in 5 ranges of

zinc concentration,

the starting material provided

6Gp1 = 6s + 2(& Both

requiring the diameter.

length was also measured with a micrometer.

etc.), it follows 6E-1 may be expressed as ,jE-1 =

87

Li-Mg

AND

Thus:

(12)

rod of composition

chemically polished to about 0.31 cm. The average of 70 to 90 micrometer measurements of the diameter * Carried out by Lucien

Pitkin,

Inc.,

New York,

N.Y.

where w, and 0,. are the frequencies

under unrelaxed

and relaxed conditions, respectively. Unfortunately, neither of these two methods for obtaining A, is suitable for present purposes without introducing appropriate corrections. The nature of these corrections will now be considered.

ACTA

88

METALLURGICA,

(a) Correction for deviation from a single rehxation

VOL.

9,

1961

(0) Correction for torsion-flexure

time

The to a

Since the Zener peak does not conform exactly

reason

coupling

for the inadequacy

of equation

relates to the fact that the bending

single time of relaxation, the rela,xation strength is greater than that obtained from the peak height using

non-isotropic rods are complicated omenon of torsion-Jlexure coupZing.

equation

which amounts to a twist when the anisotropic

(I-8).

(II-20)-that

It has been

the correction

to the deviation

shown-see

equation

for peak broadening

from a single relaxation

made by multiplication

due

time may be

by the factor El&l/T)/2.63R,

where N is the activation energy and 6(1/T) is the width of the internal friction peak at half maximum. In making this correction be calcula~d frequency

for Ag-Zn

crystals, H can

for each crystal by a~uming factor,

substituting

A,

equal

to

a constant

10r4e8se&

this value into the Arrhenius

r3) and equation

bent or, conversely,

(12)

and twisting by

of

the phenThis effect, rod is

a bend when the rod is twisted,

takes place whenever

the specimen

axis does not lie

along the (IOO), (111) or (110) directions. Because the elastic coefficients are defined under the conditions such that the specimen undergoes free distortion application

of a given stress, any constraints

with

imposed

by the conditions of the experiment (e.g. by the use of large inertia members) must be taken into account in the calculation

of these coefficients.

if the specimen

measured.(9)

general not apply.

The relaxation strength may also be measured from the change of resonant frequency in accordance with

in equations

equation

The calculations of Goen@J3) and of Hearmon(ll) show that by making the length-to-diameter ratio of

that

the

(12).

This result is not changed by the fact

relaxation

involves

more

than

a

single

vibrates

For example,

(I-3), where T is the temperature at the peak, and r = w-1. For the Li-Mg case: A and R have been

being free to undergo

“free”

in pure torsion

flexure,

equation

instead

of

(12) will in

The values of E and G that appear

(2) and (3) must be those measured under

conditions.

crystal rods oscillating

in flexure sufficiently

large, as

relaxation time. However, since AM depends upon temperature, the results must be taken from values of

in the present work, torsion-flexure

mo, and W, extrapolated to the temperature where the peak is centered. The very nearly linear nature of

play a role. Thus, Young’s modulus, E, from the resonance of single crystals is identical to the formula

the plots of frequency

employed

vs. temperature

at temperatures

outside the range of the internal friction both

the unrelaxed

extrapolation

and relaxed

ranges)

easy and accurate.

make

strengths

from the corrected peak heights.

All results for Ag-Zn

alloys

common

temperature

equation

for the temperature

tion

this

Thus, this second

method is available as a check on relaxation obtained

(II-l)-where

are normalized

to a

of 350°C with the aid of the dependence-see

T,, the critical

ordering, was estimated

from

of the various silver-zinc crystals grown is not exactly the same, each group is also normalized to the mean concentration

of that group by use of the relation A1 = As~1~/css

where c1 and cs are two concentrations,

where 1 is the length,

(13)

A, are the two corresponding values of the relaxation strength. The largest adjustment in A made with this equation was 17 per cent, but the majority of the adjustments were much less than this. The use of the above function is completely justified by the experimental results for the concentration dependence of A, to be discussed later.

p the density,

(14) T the radius and mode

of flexural vibration. For torsional vibrations of crystals of the same length but with substantial inertia members, however, correction for torsion-flexure coupling is necessary, and the rigidity modulus, G, is

given byW13) G = Q, ~(’ -x) * (1 - w) where Gi is the value of G calculated from the equation for isotropic specimens, viz, Gi = I&$/&. Here I is the axial moment attached

while A, and

namely

Q a number equal to 500.4 for the fundamental

for

Since the composition

specimens,

E = 4bpd/yr2

equa-

temperature

with sufficient accuracy

the results given in paper II.

in the case of isotropic

peak (i.e. in

coupling does not

(15)

of inertia of the members

at each end of the specimen,

and the term

< = (1 + ~~~2~12~), where M,, is the mass of the specimen, allows for the contribution of the distributed inertia of the specimen. (In the present experiments < = 1.003 and is therefore, with negligible error, employed as equal to unity.) The quantity x is given by:

x =

(s’ - *I2[ccl6

sl

+

ix36

+

cc36

-

(@I4

+

%4

+

x34)2l

P-3)

SERAPHIM

while

the quantity

(1 -

order

de~rminental

function

coefficients

ZENER

NOWICK:

AND

y)

RELAXATION

is a complicated confining

and other constants

third

the elastic

of the specimen

and

STRENGTH

IN

Ag-Zn

Li-Mg

AND

method is suitable only when the relaxation strength is relativeIy large.) In caIculating x under both relaxed

and unrelaxed

conditions

it should

the inertia members. Calculation of the term y for two specimens shows that it contributes less than

membered

4 per cent to the value of 0.

not subject to oorrections for torsion-flexure

estimates

Further,

from rough

it, was found that the eor~trib~ltion for the

remaining

specimens

of the complicated

was even less. calculations

Hence

necessary,

was dispensed with, and the equation Gz

G,(l -

in view this term

for G taken as,

x).

89

are determined In particular,

from tlexure experiments

the relaxed vaIue of S’ -

from the unrelaxed S(S -

be re-

that the values of both E and of (s’ -

s)

which

are

coupling.

s is obtainable

value and the difference

8) = (8; -

ST) -

@,I -

SJ

= --d(fWl)/dl?

(17)

(21)

The term X, which contribu~s as much as 20 per cent to G when the axis of the specimen is not close to

which follows from equation

(loo), (110) or (ill), must be included throughout the calculations. Using equations (14)-(17), G can be

of rSG-l, and therefore of AC, for orientations in which the factor x is not negligible. The corrected relaxation strength obtained in this way is generally greater

calculated

by an iteration process, i.e. substituting

approximate

value of G in equation

of x so obtained

into equation

an

(16) and the value

than hut by about 30 per cent.

(17) to obtain the next

for G. The value of E used in equation (16) is that obtained from equation (14), since no correction for torsion-flexure coupling is required for

5. RESULTS

approximation

the E-modulus. directly

from

Also, the value of s’ -

well to the relaxed unrelaxed

mod&,

samples, coupling

there

(14)-(17) moduli,

must apply

E,

E, and G,.

and

(14)-the

equally

G,, as to the

Since, for the present

is no correction

in the calculation

see equation

s is obtainable

the slope of the plot of &’ vs, l’, see

equation (2). All of the equations

for torsion-flexure

of the Young’s

measured relaxation

modulusstrength,

A E, obtained from equation (12) or from the area under the internal friction peak, is the true relaxation strength.

In the case of the shear moduIus

the relaxation

strength

(6).

The method described makes possible the calculation

calculated

Ag-Zn

(a)

Elastic coe&ients of

torsion

the

alloys

nt 350°C

Unrelaxed mod& ment

(f.c.c.)

are determined from the measure-

resonant

of variously

frequencies

oriented

in flexure

specimens

and

at tempera-

tures below that of the onset of the internal friction peak

For present purposes

lated to 350°C to obtain this

temperature.

(14)-(17) coupling.

In

such data are extrapo-

the unrelaxed this

calculation

moduli

are utilized to correct for torsion-flexure Fig. 1 shows the linear plots of E,-l and

however,

by these methods

may be given the symbol A,(, representing the correct relaxation strength for an isotropic specimen (or a specimen in one of the three special orientations). Thus, dG,-1 = Cri--l represents

the

relaxation

0,.-i of

= C,.-fA& the

uncorrected

(13) reci-

procal modulus. To obtain the corrected relaxation 6G-i, one must calculate both corrected moduli, 0, and G,., from an equation of the type (17) by the iteration method already discussed. In this procedure one uses for G, the value Gui = 11&,;/n@ based on equation (15). On the other hand, obtajned from equation (18) as %-l

= G,<-‘(I

+ A,$

(19) GVi is

(20)

(Gvi may also be obtained from W, using equation (19) with the subscript r substituted for u. This latter

0

at

equations

I

I

1

0.1

0.2

0.3

FIG. 1. Dependence of reciprocal unrelaxed moduli on orientation for the alloy Ag-26 at. o/0 Zn at 350°C.

ACTA

90

GU-r

vs.

the

orientation

~ETALL~R~ICA,

function, I’, The EW-l curve,

VOL.

for the which is

on the orientation

function

1961 Temperature, “C

300”

26 at.% alloy at 350%. the more reliable one due to the absence of corrections for torsion-flexure coupling, shows the linear dependence

3,

.05 I

I

I

4OGQ I ,

I

I

500” I I

,

I

I’, as required by

(2).

The intercept at I’ = 0 gives (8,’ $s,“)/3 = 39.5 x lo-‘3 cms/dyn and that at P = 0.333 gives (s,, + a,“)/3 = 10.0 x lo-l3 cm2/dyn. equation

In the present circumstances fraction

of the effective

where s,* is a small

reciprocal

moduli and there-

fore difficult to obtain with any precision measurements,

from these

it is va.luable to note that a sufficiently from the of s,” can be extrapolated

reliable value results of Bacon and Smith(i4! on more dilute Ag-Zn alloys.

Hence,

with the use of sU’ obtained

in this

way, and the two intercepts

of the Es-l curve quoted

above,

and

the

quantities

s,

sU’ are

completely

determined. From equation (3), the line for G,-r vs. I’ can then be drawn witli.out reference to the data points. The straight line for the curve of GU-i vs. I’ in Fig. 1 is obtained

in this way.

at 350°C obtained Ag-Zn

The best values of s, and 8,’

by this procedure

alloy concentrations

for the various

are summa~zed

in Table

1, together with the values of SUNdeduced by extrapolation from Bacon and Smith’s data. A more complete

report

concerning

the

concentration

-

and

1000/1

Fxa. 2. Internal friction peaks for flexural oscillation of a series of orystals of Ag-Zn alloy all of very newly (110) orientation. The number labeling each curve IS the atomic per cent of Zn.

temperature dependence of these coefficients for Ag-Zn f.c.c. solid solutions will be published separately.

permit accurate measurement A plot of internal friction of

Ag-Zn

essentially

of

zinc

concentrations,

the same orientation,

An important ground

various

peaks for single crystals but

of

is shown in Fig. 2.*

feature of this figure is the low back-

internal friction

for polycrystalline

at high temperatures,

specimens

the beginning

boundary peak would normally appear. Because of this feature, and the fact that the values of relaxation strength

are large

(for concentrations

20 at. y0 Zn), the relaxation by the two

independent

greater

than

strength can be calculated methods

discussed

in the

previous section: (a) from the height of the peak and (b) from the temperature variation of the resonant frequency. In both cases appropriate corrections, described in Section 4, must be made.

As an example

of the excellent agreement between the two methods of measuring the relaxation strength, the deviation from the mean of the two values of AE obtained for twelve specimens in flexure, averages to 1.9 per cent. For the three lower ranges of concentration it was fouud

that the relaxation

was not strong enough to

* The shift in peak position with concentration is due to the strong dependence of activation energy on concentration.‘3~

from the change

culated from the height of the peak, by the method described

where of grain

of A,

in modulus. Therefore, all values of A, to be reported for these lower concentrations will have been calin Section 4.

The most complete carried

out

on

the

set of measurements

have been

groups

containing

of

crystals

30 at. y0 Zn and 26 at.% Zn. The orientation dependence of the Zener relaxations 6E-1 and 6G-1 are shown in Fig. 3 for the crystals of 30 at.76 Zn and in Fig. 4 for

the

crystals

of 26 at.:/, Zn.

value of 6E-1 for a given multiplying the measured (computed

in accordance

by the reciprocal

experimental

with the previous

of the unrelaxed

of the same crystal,

The

crystal was obtained relaxation strength Young’s

i.e. 6E-l == E,-iAE.

by A,

section) modulus A similar

procedure was used for 6G-l. Equations (6) and (9) of the formal theory predict that SEW and 6G-i are linear with r and that the slope 6G-r is twice that for SE-l but of opposite sign. The straight lines drawn in Figs. 3 and 4 fit this requirement and at the same time are in excellent agreement with the experimental points. Since there are no questionable assumptions involved in the formal theory, such

SERAPHIM

AND

I

NOWICK:

I

ZENER

RELAXATION

1

I

STRENGTH

IN

Ag-Zn

Li-Mg

AND

TABLE 1. Unrelaxed elastic coefficients for Ag-Zn f.c.c. solutions at 350°C (in units of lo-*” cma/dyn)

,

At. y0 Zn

8”

30.3 26.1 16.6 10.6 3.7

27.0 26.0 25.5 25.0 24.5

* Obtained Smithu4’.

122 115 103 94 85

by extrapolation

0.1

4.0*

/ I

0.2

68

8s’

30.3 26.1 16.6 10.6 3.7

0.60 0.50 0.20* 0.07* 0.01*

22.5 17.1 5.5 2.5 0.3

0.3

FIG. 3. Orientation dependence of the relaxations of the reciprocal moduli for the alloy Ag-30.

agreement

ye Zn at 350° C.

must be found if the experimental

* Values extrapolated using equation (13).

results

are correct. The results for each composition are best described in terms of the three relaxations 68, 6s’ and 68” which can, in principle, be computed from the intercepts at I’ = 0 and I’ = 0.333 with the aid of equations (6) and (9). Unfortunately in this particular application, because of the small values of the intercepts

6E-1 (111) = (6s + &“)/3 and 6G-l (lOO) = &s,

it turns out that the results within experimental

can be accounted

be combined

from

_____ I

)_zinc

and substituted

and

for Ag-Zn

65” ____ < 0.3 5 0.1 -

higher

-

concentrations

into equations

and (11) to obtain the “formal relaxation strength as a function

(10)

theory” curves for of r. These curves

for the 30 and 26 at.% Zn alloys with the experimental

3.5*

are plotted

along

points in Figs. 5 and 6, respec-

tively,

The results show a strong anisotropy

AE and

A,. In terms of anisotropy ratios, defined as

in both

for

error with 6s” = 0. Consequently,

only an upper limit for 6s” can be estimated.

These

results for the two alloy compositions under consideration, are presented in the first two lines of Table 2. The data of Tables 1 and 2 for these compositions

0

now

i:: ,.,i

from the data of Bacon

At. y0 Zn

r

solid

n 8%

%

TABLE 2. Relaxations of the elastic coefficients f.c.c. solid solutions at 35O’C (in units of lo-r3 cm2/dyn)

-0

91

0.2

0.1

0.20 7

f

I

I

I 0.1

I 0.2

I 0.3

may

d.3

r FIG. 4. Orientation deuendence of the relaxations of the reciprocal moduli for the alloy Ag-26 at.% Zn at 350°C.

0

r

FIG. 5. Orientation dependence of the relaxation strengths for the alloy Ag-30 at. y0 Zn at 350” C.

_

ACTA

92

METALLURGICA,

VOL.

9,

1961

in order to get a better idea of the dependence

of the

relaxation

strength on composition as well as on Since it was not possible to obtain 6s” orientation. from

the

intercepts

~YE-~(lll)

and

6G-1 (100)

in

Figs. 2 and 3, and since these intercepts become even smaller with decreasing zinc concentration, there is little value in including lower composition

a” 8 a”

alloy compositions present experiments the two relaxation I

Ss” in the calculations

ranges.

coefficients,

these are not complicated coupling correction, which

8s’ and as, which can

I

I

I

0. I

0.2

0.3

by the torsion-flexure is difficult to compute

exactly-see Section 4(c). The relaxations of Young’s

\

0

for

below 26 at.“/: Zn are for the adequately described in terms of

then be determined from experiments in flexure alone. The advantage of using flexure experiments is that

.

I

in the

Thus the relaxations

r

modulus

for the com-

I

I

1

FIG. 6. Orientation dependence of the relaxation strengths for the alloy Ag-26 at.% Zn at 350°C.

the results obtained

Atom % Zn

for the two alloys are given in

\

Table 3. The relaxation coefficients

the form: given in Tables relaxation

30.3

strengths A,, As, and A,- for the s, s’ and a”, are defined by equations of the quantities A, = &s/s. Combining 1 and 2 permits

strengths,

given

calculation

in Table

3.

of these The

large

ratio of the relaxation strength for the al-shear to that for the s-shear provides another measure of the strong anisotropy

of the relaxation

phenomenon.

The reason for the large uncertainty

in A, in spite

e.6

-nArb\

of the fact that it may be of the same order of magnitude as A,, is the small dilatational strain that actually exists in a sample in tension. For example, for tension of a (111) bution the

oriented

sample

the dilatational

to the tensile strain is about

total;

for

other

orientations

contri-

one seventh the ratio

is still

smaller. this

of

The small value of sU” also contributes It is evident, therefore, that situation.

to an

accurate value of the dilatational relaxation strength for alpha Ag-Zn, is best obtained from an experiment in which the sample is subjected

to purely hydrostatic

----_-_

3.7

____ 0.1

0

alloys of various compositions.

plete set of Ag-Zn

crystals are shown in Fig. 7.

data very near the (111) orientation below 16.6 at.74 Zn, the relaxations extrapolated from The extrapolation

At. % Zn

(

RE

/

&

1

A,

1 A,’

1

A,”

for

In

view of the small values of dE-l (111) and the lack of

these alloys are best calculated

at 350°C

0.3

FIG. 7.Orientation dependence of 6E-'for Ag-Zn

Now that the results for the 30 and 26 at.% Zn alloys have been examined, it is appropriate to consider measurements at lower zinc concentrations strengths and anistropies two a Ag-Zn solid solutions

oe

r

stress.

TABLE 3. Relaxation

I 0.2

for specimens 6E-1(lll) for

from values of AEclll>

the higher zinc concentrations. was performed with the aid of equation (13), which will be verified later. With these intercepts obtained in this way and the experimental data for specimens containing 3.7, 10.6 and 16.6 at.% Zn, t)he large intercepts (at I? = 0) can be determined. In this way the data for the lowest three concentration alloys quoted in Table 2 are obtained. The relaxation strengths, AZ, are given as a function of I’ in Fig. 8.

SERAPHIM

AND

ZENER

NOWICK:

RELAXATION

STRENGTH

-where

D(r)

Atom %

Zn

given

-I

Ag-Zn

93

AN11 Li-Mg

depends on I’ but not on c-suggests

proportionality 0.20 c

IN

between

orientation

the relaxation

and the

quantity

a

strength at a E,-ic2(1

-

c)~.

On the other hand, the results of paper II suggest as (T - T&l, that AM varies with temperature where

T, is

itself a strong function

Accordingly,

a” 0.10 -

a

plot

of concentration.

(T - T,) AEcu,,,, against

of

E,-k2(1 - c)~is plotted

16.6

presented in Fig. 9, curve A. The points were taken from the curves in Fig. 8

and the values of

0.05 -

results

show

linear relation 0.10

0

r

FIG. 8. Orientation dependence of AE for Ag-Zn of various compositions.

Here again the curves drawn fit equation the parameters given in Tables 1 and 2.

predicted

pair-reorientation

alloys

dependence

of

Fig. 9 as curve (10) with

Finally,

theory-see

strength to concentrations

30.3 at. “/ alloy.

empirical

one cannot

= M,-~P~B(

as a test of the Zener equation

be used to distinguish justify

as obtained

of AE is, of course, also evident in Fig. 2, for crystals near the (110) orientation.

A,

a c2 dependence

in curve

viewpoint,

as high as 30 at.?< solute,

B. one

Finally, may

plot

from

differences

c* E,

c2.

strength

for

It should be noted that

(13) was used in the analysis

of the data to determine

A3(i1i>

for the lower zinc

alloys, did not significantly prejudice the results obtained in favor of the cs dependence. This statement

c*(I-C)*E,’ (cm*/dyne

I

in composition.

the fact, that equation

* Actually equation (I-14) only agrees with equation (23) if the functionf(p, c) which appears in (I-14) is independent of c. This condition is satisfied when the order parameter p is small.

8

a purely

AEclO,,> vs.

with no more scatter than curve B, Fig. 9. This result is then consistent with equation (13), which

(23)

6

theory,

of the relaxation

Such a plot, though not shown, gives a linear relation

which

c)s

(1-12)-the

the two theories.

has been used to correct the relaxation r)c2( 1 -

the

of Le Claire

In fact, even in terms of the pair-reorientation

dependence

see (I-14),

The

this plot is also linear over the region of investigation. It is clear, therefore, that the concentration depend-

A E(lOO) for the 10.6 at .yO Zn alloy is a factor of 7.5 lower than the value of the same parameter for the

equation,

error,

(T - Te)AE(icO) on E,c2is given in B. Within the experimental precision,

ence cannot

Le Claire and Lomer’s predicts that*

experimental

by the theory

One of the most striking features of Fig. 8 is the strong concentration dependence of AE, particularly as evident at r = 0. For example, the value of

The strong concentration

Fig. 3 of paper II.

within

and Lomer is obeyed.

0.30

0.20

T, from

that,

IO

X lot41 12

14

16

I

I

2

3

(dynes /cm2 ‘X la’“)

FIG. 9. The concentration dependence of AE for (100) oriented crystals of cc Ag-Zn. Curve A : (2’ - Te) . Aa plotted against c2( 1 - c)~E,-‘. Curve B: (T - !P’,) . Arc plotted against c2E,. In both cases, AE and E, am values for the (100) orientation.

94

is based on the large anisotropy

ACTA

METALLUR~ICA,

of A,,

due to which

the concentration dependence of AEcu,,,> is hardly affected by the manner in which the best value of A E(l11) is selected, within reasonable limits. Also, the previous use of equation (13) to normalize the

9,

VOL.

1961

TABLE 4. Elastic coefficients, relaxations and relaxation strengths for b.c.c. Li-Mg solid solution at 115°C ___.~ .-... -

-

‘--

38

tions

to the data were always

cannot in any way have affected Fig. 9. (b) Li-Mg Internal

friction

small,

so that they

peaks for flexure and torsion of a

Tempemture

(F)

too

the same for pure lithium.05

The important feature of this high value of a” is that it suggests the possible detection of a dilatational

oxis at>

relaxation

which until now has escaped measurement.

The relaxations

“I!

5 E 0 s

as s for this

this is not surprising since the relative magni-

tudes are also about 150

-8

s F .o c

5,” is of the same order of magnitude alloy;

10

-&*

4.65

(6.c.c.) aZEoy

50

Specimen

I

0.17

Is cmz/dyn

the result found in

(111) oriented single crystal of Li-57 at. o/o Mg are shown in Fig. 10. These peaks are small compared

25

34

179 x 10

relaxation strengths for a group of crystals to those for a mean concentration is justified. These correc-

_

----

torsion 6

are plotted

dE-1 and &G-l for flexure and for in Fig. 11.

The line for 6G-’

drawn with twice the slope of dE-l, theory.

4

tory.

The agreement From

with experiment

the intercepts

is

to fit the formal is satisfac-

one can calculate

as, 6s’

and 6s”. The results, presented in Table 4, show a significant dilatational relaxation contribution, 68,

2

for the present alloy, in contrast to the case of Ag-Zn. 3.4

3.2

3.0

2.8

2.6

2.2

2.4

The relaxation strengths A, and A, are shown in Fig. 12 for these Li-Mg single crystals. The results

1000

--TIO. Internal friction peaks of a single crystal of Li-57 at.% Mg alloy oscillating in torsion and in flexural Torsional frequency: 490 c/s; flexure. frequency: 1850 c/s. Fra.

to the peaks for the high concentration

Ag-Zn

consequently,

is much more

the relaxation

strength

alloys;

follow

the pattern

alloys,

where the maximum

shown

previously relaxation

by the Ag-Zn strength

AE

is in the (100) orientation, while that of AC is in (111). The anisotropy ratios as defined in equation (22) are

precisely obtained from peak heights than from the difference between relaxed and unrelaxed moduli. The peak for flexure

in Fig.

10 occurs

at a higher

temperature than that for torsion only because of the higher frequency of the flexural oscillations. The difference

in peak height

immediately

indicates

the

existence of anisotropy of the same type as that which was found for the Ag-Zn alloys. Complete best obtained

information

on relaxation

by using crystals

anisotropy

of different

is

orienta-

tions. In t-he same manner as for the Ag-Zn alloys, the principal unrelaxed elastic c~~oients s,, s,’ and SuR may be obtained from the variation of E,-l and GUal with I’. In the present alloy, because of the relatively high compressibility, there is no difficulty in determining sUfl in this way. Table 4 summarizes the data for the elastic coefficients obtained for the Li-57 at.:!,,- Mg- alloy” at 115’C. It is to be noted that

5

01

0

I

0. I

I

1

0.2

0.3

r

Fra. 11. Orientation dependence of the relaxations 6E-' and 6ff-’ for the alloy LX7 at.% Mg at 115°C.

SERAPHIM

AND

ZENER

NOWICK:

RELAXATION

STRENGTH

which

this ratio

shown.*

IN

Ag-Zn

Li-Mg

95

the theory

is

AND

is calculated

from

Using the present values of elastic constants

brings the th~reti~al Ag-30%

Zn alloy.

correctly

estimate of

R, up to 2.1 for the

It is clear that although the theory

predicts

the

direction

of

the

anisotropy

(i.e.2 AC\,<,,,, < Aaciix,), it is unable to show why the anisotropy is actually so large. The source of the theoretical prediction small anisotropy of

the lattice structure;

there are a sufficient

I

,

I

I

0

0.1

a2

a3

I

= 2.8 and R,

is in the same direction somewhat

lower

intercepts

in Fig.

= 5.2.

favorably roughly

Thus the anisotropy

as for the f.c.c.

in magnitude.

The

alloy

rather

11, as well as the fact

crystals used had orientations

but large

that the

close to ( 100) and ( 111),

provide here comparatively precise values of A,, A, and A,),. These values are also given in Table 4. It is noteworthy per cent.

number

of nearest

neighbors

matters in what crystal direction the stress is applied, since there will always be both favorably and un-

i

R,

in terms

namely, in the f.c.c. structure

orientations (i.e. six) as to produce near isotropy in the relaxation behavior. Stated differently, it hardly

Fm. 12. Orientation dependence of the r&x&ion strengths for the alloy Li-57 at.% Mg at 115°C.

here:

of relatively

may be seen qualitatively

that the uncertainty .The presence

in ASS is only

of a dilatational

effect is thus clearly demonstrated

17

relaxation

for the first time.

oriented

nearest

neighbor

positions

in

equal numbers.

At this point let us consider the case of b.c.c. Li-Mg. Here both the pair-reorientation and direetional

ordering

theories

that the relaxation be zero. readily

make a striking

prediction:

strength for the s’ shear A, will

The reason for this prediction as follows.

In the b.c.c.

may be seen

lattice

the (100)

direction makes equal angles with all four nearest neighbor directions, which lie along the cube diagonals. Therefore,

application

of a tensile stress along (100)

must lead to a relaxation

strength

except for the dilatational

relaxation.

AE which is zero From equation

(6) it is clear that in this case 6s’ -=: 0 and therefore 6. DISCUSSION

The existence

of a dilatational

effect (as first predicted the case of the Li-Mg pair-reorientation

by Le Claire and Lomer)

in

alloy, means that the original

theory

the Zener relaxation

A, = 0. or bulk relaxation

cannot adequately

phenomenon.

describe-

Nevertheless,

this

Correspondingly,

maximum 6s $ 0.

value Further,

for

interactions.

directional

(111)

The

its since will

only nearest

pair-reorientation

a bulk relaxation,

shows that this prediction

the

on

and

ordering theories are special cases of this

general type of theory. Examination of the experimentaf

to modify

take

orientation,

which considers

fact does not eliminate the pair-reorientation model from consideration. To enable this model to predict it is only necessary

will

it is clear that this prediction

come from any theory neighbor

AE

the

results for Li-Mg

of A, = 0 is not obeyed.

theory to take into account the fact that there will be

Instead,

a change in the number of solute pairs in the presence

A SI> A,, and that AE is a maximum, not a minimum, in the (100) direction. At this stage an argument may be given suggesting

of hydrostatic pressure (or tension). For the case of pure shear stress, e.g. in the torsion of wire samples, the original change.

pair-reorientation

model

requires

no

observed in both the f.c.c. and b.c.c. cases.

Consider first the results for the f.c.c. Ag-Zn solid solutions. IIere we have found anisotropy ratios R, and R, of about 9 and 7, respectively. By comparison, LeClaire and LomeG@ estimate a value of about 1.7 for R, of Ag-Zn, based on their directional order theory and data for elastic constants then available. In the Appendix

it would

conclusions

The major result of the present work is the strong anisotropy

that

we have found

to the present paper, the manner by

(as for the f.c.c.

be dangerous

to draw

case) that

any

general

from the Li-Mg work on the grounds that

this alloy is not typical

of those usually

considered,

in that the atoms Li and Mg arc not of the “hard core” type characteristic of the noble metal atoms with their completed d-shells. This argument may be answered in two ways. First, as already pointed out above, there is good evidence for a discrepancy between theory and experiment * The anisotropy predicted theory is worked out by Berry

in the case of the f.o.c. by the pair-reorientation in paper IV of this series.

ACTA

96

METALLURGICA,

VOL.

9,

1961

Ag-Zn solid solution, which is a noble metal type alloy. Here the measured anisotropy, though not in the

higher than the value which would have been obtained

wrong direction,

as the others.

is considerably

greater than we have

if these specimens had been run at the same frequency

a right to expect if only nearest neighbor interactions

small amount

are taken into account.

produce,

of Artman

on o-brass,

Secondly,

which is a b.c.c. solid solution

of the noble metal type. as a function

Fig. 13 shows AE and SE-I

of r calculated

The results appear

there is the work

much

from Artman’s

like those

data.c5)

of the present

tropy

One of the two specimens of a-phase

if anything,

present,

a high result.

in @-brass, therefore,

also had a

which would

is possibly

even greater

than that obtained from Fig. 13. It is concluded

somewhat

that p-brass

possesses

relaxation

as Li-Mg. Comparing

all the results for the anisotropy

strength

at (111).

The anisotropy

ratio,

= 3.8, is slightly greater than the corresponding

ratio for Li-Mg. In Artman’s own discussion take into account

of his work he did not

the two specimens

close to (111)

that,

in spite of the complications of ordering and formation of a second phase which make the data of Fig. 13

investigation, showing again a maximum relaxation strength for tension at (100) and a much smaller R,

also

The true aniso-

inexact,

these data without the same type

a doubt

show

of anisotropy with

theoretical predictions, it is clear that any theory based solely on nearest-neighbor relationships is unable to explain the observations. At this stage, it is useful to make

a completely

which are so necessary for the analysis. Instead he placed more faith in the four specimens between

ad hoc assumption,

I’ = 0.06 and l7 = 0.16 which led only to the con-

direction in which one must look to find an improved

clusion that AE
theory.

strength

phenomenon.

appears

isotropic

for these four specimens

only because they fall in range of I’ where the dependence upon orientation

is small.

The two specimens

as a means

The assumption in

dealing

to each other need be

with

process, and not the reorientation

shifted to a higher temperature,

lie along the (100) direction.

specimen.

the long-range

order of the

(The peaks for all the specimens

in Art-

man’s work fell below the critical temperature, range where the long-range complete.)

Since

the

in a

order is about four fifths

Zener

peak

increases

with

Zener

the reorientation of next-nearest atom pairs are assumed to contribute pairs.

shift will have decreased

the

relaxation

Thus, in terms of the pair model, only

which Artman did not take into account were omitted because they were oscillated at a higher frequency than the others. As a result, the peak must have been and this temperature

the

is that only atoms in the next-

nearest neighbor relationship considered

of indicating

neighbor solute to the relaxation

of nearest neighbor

It should next be noted tha,t in both the f.c.c.

and the b.c.c. lattices the next-nearest (111) direction

neighbor pairs

Accordingly,

since the

makes equal angles with these next-

nearest neighbor pairs, it follows, according to our assumption, that AE(iil) should be zero, except for the bulk relaxation, or, from equation (6), that 6s = 0. On this basis the prediction

would

be made that in

decreasing order of an alloy,(17) the observed relaxation

both the f.c.c. and the b.c.c. lattices there should be a

strengths for these two specimens should be somewhat

strong anisotropy

I

p-

BRASS

(AFTER

in AE going from a maximum

value

in the (100) direction to a near-zero value in the (111) direction. Certainly this prediction is more nearly in accord

R. ARTMAN)

from

with the observations theories

neighbor

based

relationships.

ad h,oc assumption neighbor

on

relationship

to the relaxation

than that which came

considering

only

nearest

If now we relax the above

to state

that

the next-nearest

makes the major

contribution

process but that some contribution

also comes from nearest neighbor pairs, the observations may be explained rather well. It is even possible to understand

why the anisotropy

ratio is larger in

the f.c.c. alloys than in the b.c.c. alloys, since the anisotropy of the nearest neighbor contribution is in

FIG. 13. Orientation dependence of 6%’ B-brass. (After Artman’5’.)

and of

AE for

the same direction as that of the next-nearest neighbor contribution for the f.c.c. case, but in the opposite direction for the b.c.c. case. Although

it is possible to explain the results in terms

SERAPHIM

of the assumption

of the dominance

of next-nearest

neighbor

behavior

that

over

ZENER

NOWICK:

AND

RELAXATION

of the contribution

atom pairs to the relaxation

of nearest

neighbor

pairs,

the

reason why this should be the case is not clear. One possible explanation is that the strain field about a nearest

neighbor

solute

pair is much

more

nearly

isotropic than that about a next-nearest neighbor pair. Thus, the effect of pair-reorientation in producing anelastic

strain is greater for the former than for

the latter case. This question of further investigation. It may be concluded

is certainly

deserving

that the old pair-reorientation

theory of Zener may be utilized successfully pret

the the~od~amics

particularly

of the Zener

at low concentrations,

theory is modified

provided

in two ways.

pressure

dilatational

relaxation

in order

that the

First, account

be taken of the change in the number hydrostatic

to inter-

relaxation,

to

account

phenomenon,

must

of pairs with for

the

Secondly,

at-

tention should be directed to pairs in next-nearest neighbor configurations more than to those in nearest neighbor

configurations.

for low concentrations. ACKNOWLEDGMENTS

This work was supported

by the Office of Ordnance

Research, U.S. Army. This paper represents part of a dissertation submitted by D. P. Seraphim in partial fulfillment

of

the

requirements

Doctor of Engineering Yale University.

IN

Ag-Zn

AND

Li-Mg

97

4. B.G. CHILDS andA.D.L~C~~1~~,ActnMet.2,718 (1954). R. A. ARTMAN, J. Appl. Phys. 23, 475 (1952). :: A.S. NowIcK~~~R.J.SLADEK,AC~~M~~. 1,131(1953). 7. A. E. ROSWELL and A. 8. NOWICK, J. Met., N.Y. 5, 1269 (1953). of Metals. University 8. C. ZENER, ~~8~~~~~~a& AneZasticilgt of Chicago Press, Chicago (1948). 9. D. P. SERAPEIM, Tpans. A9r~. In&. Min. (~~e~~2.) Esgrs. 218, 485 (1960). 10. T. S. K1, J. Appl. Phys. 20, 1226 (1949). R. F. S. HEARNION, Rev. Mod. Phys. 18, 409 (1946). :x. E. GOENS, Ann. Phys., Lpz. 11, 649 (1931). 13: E. GOENS, Ann. Phys., Lpz. 15, 455 (1932). Acta Met. 4, 337 (1956). 14. R. BACONand C. S. SMITH, 15. H. C. NASH and C. S. SMITH,J. Phys. Chem. Solids 9, 113 (1959). 16. A. D. LECLAIRE and W. M. LOMER, Actn Met. 2, 731 (1954). 17. J. JULAY and C. WERT, Acta Met. 4, 627 (1956). APPENDIX To estimate t,he anisotropy of the relaxation strength, L&l&m and Lomer~‘~~make use of the fact that the orientation dependence of the relaxation strength, viz. the quantity DM( I?) of equation (23), is given in the directional ordering theory by D,( I’) -= Ccc/M, (Al) where C is a constant independent of orientation, and TVis a dimensionless constant which is given by

In spite of these changes,

the pair-reorientation theory still explains why the relaxation strength varies as the square of the solute concentration

STRENGTH

for

the

degree

of

in the School of Engineering

at

The authors are indebted to B. S. Berry for helpful discussions, and to F. G. Fumi for reading the manuscript. REFERENCES 1. A. S. NOWICE and D. P. SEIRAPB[IM, Acta Met., 9,40 (lQ61). 2. C. Y. LI and A. S. N~WICK, Acta Met., 9, 49 (1961). 3. A. S. NOWICK, Phys. Rev. 88, 925 (1952).

Here a is the irrteratomic spacing at zero strain, and a, the in~ra~mic spacing under strain e in the various nearest neighbor dire&ions (i = I,2 . . . . , z/2). LeClaire and Lamer show that, for the f.c.c. lattice, a = l/2 for the s-shear and o = l/4 for the s’shear. Since 6s = sA, and 6s’ = .s’A,, we obtain, by inserting these values into equation (Al),

Sd -. = _ad2 68

2s,=

(83)

From this result, it follows from equations (22) and (1 I) that

For Ag-30 at.% Zn, the present results (Table if give %4fl% = 4.5, or Rg = 2.1. This value differs from the value Rs = 1.7 given by LeClaire and Lamer due to the more accurate values of elastic constants now available. Equation (-44) shows that the LeClaire-Lamer theory predicts a value of Ra with almost no uncertainty (i.e. to better than 10 per cent), in view of the acouracy of the measured elastic constants. A serious quantitative discrepancy between the predicted value and experiment cannot hhen be attributed to experimental error.