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.