Crystallography of martensitic transformation ′ith long period stacking order

CRYSTALLOGRAPHY OF MARTENSITIC TRANSFORMATION WITH LONG PERIOD STACKING ORDER* H.

KUBOt

and

K.

HIRANOS

On the basis of the recent experimental observations of the marten&tie structure exhibiting long period stacking orders, the orystallography of mrtrtensitic transformation has been developed which enables the features of transformation mvolving the structures of this kind to be predicted. The point of the theory is to specify the lattiae deformation matrix as a product of three matrices which represent pure deformation, homogeneous shear and shuffling. The application of the theory to the martensitic transformation in Cu-Zn alloy hw shown to give results in satisfactory agreement with the available experimental dsta. The theory predicts &line&rmlationship bet,weenthe two parameters CL and /l which represent staeking fault probabilities. Electron diffraction studies have been carried out to eonfirm this relationship. CRISTALLOGRAPHIE DES TRANSFORMATIONS MARTENSITIQUES ORDRE D’EMPILEMENT DE GRANDE PERIODE

PRESENTANT

UN

La cristallographie de la transformation martensitique, permettant de prevoir les caraet&istiques des structures martensitiques presentant un ordre d’empiiement de grande periode, 8 et& &abbe a partir dea observations experimentales recentes effet,ubs sur ~88 structures. L’essentiel de cette theorie eat d’exprimer la m&rice representant la deformation du r&eau par un produit de trois matrices rep& sentant la deformation pure, le cisaillement homogene et le glissement. L’application de la theorie a le transformation martensitique d’un alliage Cu-Zn donne des result&s en accord satisfaisant avec les result&s experimentaux. La theorie prevoit une relation lineaim entre les deux parametres CLet fi qui representent les probabilites de faute d’empilement. Des etudes de diffraction Blectronique ont ete effect&es pour conflrmer cette relation. DIE

KRISTALLOGRAPHIE DER MARTENSITISCHEN UMWANDLUNG PERIODISCHER STAPELORDNUNG

MIT LANG-

Ausgehend von neueren experimentellen Beobachtungen der Martensitstruktur mit langperiodischer Stapelordnung wird die Kristallographie der mertensitischen Umwandlung entwiokelt, die diese Art Strukturen vorhersagt. Kern der Theorie ist die Darstellung der Gitterverformungsmatrix als Prodnkt dreier Matrizen: der reinen Deformation der homogenen Seherung und der Versohiebung. Die Anwendung der Theorie auf die martensitrsche Umwandlung in Cu-Zn-Legierungen liefert befriedigende %erseinstimmung mit experimentellen Ergebnissen. Die Theorie sagt einen linearen Zusammenhang zwischen den zwei Parametern a und /?, die fur die Stapelfehlerhaiifigkeit stehen, voraus. Zur Bestatigung dieser Beziehung wurden Elektronenbeugungsexperimente durchgeftihrt.

1. INTRODUCTION

A new group of crystal structures, in which the stacking order of the close packed planes is int’ermediate between those of fee and hcp has been found in numerous noble metal alloys(r*~ and transition metal alloys.(i) The existence of such structures has been recognized also in martensites of nonferrous allovs such as Cu-Al,uJ-12) Au-Cd(isJ4) and CuZn.(i5*r6) These may be called shear structures or structures with long period stacking order. In view of the theory of the martensite crystallography, the complexity of these structures has prevented us from estimating the matrix B which represents the lattice deformation associated with the martensitic transformation. It will be elucidated that the lattice defo~ation is composed of three elementary parts, i.e. pure distortion, homogeneous shear and shuffling. Hence the lattice deformation * Received April 9, 1973. t Sow at: Institute of Scientific and Industrial Research, Osaka Universtty, Suita, Osaka, Japan. 2 Department of Materials Science, Faculty of Engineering, Tohoku University, Sendai, Japan. ACTA

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of the martensitic transformation with long period stacking order can be specified by the multiplication of three matrices which represent those three elementary parts, The theory is applicable to any martensitic transformation with long period stacking order. The transformation in Cu-Zn alloy from the B2 to 9R structure which has not the ideal close packed structure has been adopted as an example to apply t,he theory. The theory predicts a particular relationship between u and /3 which are defined as the continuing probabilities of the cubic type and the hexagonal type stacking, respectively.‘i7-la) Electrondiffraction study on single crystal specimens of Cu-Zn alloy has been carried out to confirm the relationship. 2. THEORETICAL

TREATMENTS

2.1 Geometrical considerations The structure of the parent phase /3, in Cu-Zn alloy which has an ordered structure of B2 type can be 1669

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OZn

FIG. 1. Structure of the parentphase (B2 type).

considered to be an alternate stacking of the (110),1 planes A, and B, as shown in Fig. 1. If the planes of A, and B, are transformed to produce the close packed planes of the QR structure, they have to undergo the following deformations. (i) Compression along the [001],1 direction and expansion along the [flO]s, and [11O]P1directions (cf. Fig. 2). In order to make the perfect QR structure, the sequence of close packed planes has to be arranged as in the following artificial way. (ii) Shear on every (llO)@, planes in the direction of [liOla,, i.e. homogeneous shear (cf. Fig. 3 (b)). (iii) Reverse shear on every third (11O),1 planes in the direction of [ilOJPl, i.e. shuffling (cf. Fig. 3 (c)). Thus, the lattice deformation of the martensit,ic transformation from the B2 to the QR structure can be regarded as the composition of three elementary parts (or processes) as mentioned above. Figure 4 illustrates the change in a unit sphere abed by these processes. The unit, sphere undergoes t,he pure deformation by t’he process (i) to produce the first ellipsoid u’b’c’d’ and is further converted into t,he ellipsoid a”b”c”d” by the process (ii) + (iii), i.e. homogeneous shear plus shuffling. The amount of inhomogeneous shear F,is determined so that the ellipsoid a”b”c”d” is in contact with the unit sphere. In the case of the QR structure in Cu-Zn

FIG. 2. Three types of the basal planes of martensites with long perrod stacking order, transformed from the close packed plane Al of B2 type ordered lattice.

(a) Distorted

(b)f c c

PI

(C)SR(/%)

Fro. 3. (a) Stacking of distorted (llO)& planes viewed from [OOllp, direction. (b) f.c.c. structure produced b\ homogeneous shear in the [lIO]p, direction on distorted (llO)pl planes. (c) 9R structure produced by shuflling at every third layer on the basal planes of the f.c.c. structure.

alloy, the inhomogeneous shear (slip) occurs on the same crystallographic system as that in t’he processes (ii) and (iii).

Then, the possible habit, planes relative

to the product intersection

lattice

are given by the planes of the

of the shape ellipsoid (denoted by a dot-

dash-line in Fig. 4) and the unit sphere which touches each other in one direction

in the plane of inhomo-

geneous shear. Figure

5

shows

the

viewed from the direction

stereographic

projection

of the principal

axis of the

lattice deformation, f,. Since the arcs A”B”C” and D”E”F” are the loci of intersect’ions of the unit

sphere and the ellipsoid a”b”c”d”, they produce the final cones of the unextended line of the lattice deformat,ion, while A’B’C’ and D’E’F’ produce the initial cones of unextended line corresponding to A”B”C” and D”E”F”, respectively. In Fig. 5, Ii, is the inhomogeneous shear planes and Ii, and Ii,’ are the planes which make angles of 90’ f 13with the

FIG. 4. Section in the mane of homoaeneous shear and shuffling. A umt sphere abed is distorted into the first ellipsoid a’b’c’d’ by pure deformation and u”b”c”d” by lattice deformation for the 9R structure. The dot-dashline indicates shape ellipsoid.

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matrix H, representing the process (ii) as H, = I + h&m,

(2)

where I is a unit matrix and h1 represents the magnitude of the homogeneous shear. Since the matrix H, representing the process (iii) can be given in t’he similar form as H,, the magnit,ude of the shufaing h,, the direction 1, and the plane q’ of the shuffling are defined according as h,, l1 and m,‘. Provided that the homogeneous shear and shuffling take place in an as yet untransformed cubic phase, t.he lattice deformation may be represented by B, = TCT’H$I,

FIG. 5. Stereographic analysis of martensitio transformstion with the 9R structure in Cu-Zn alloy.

(3)

If Be is a nonsingular matrix, it is factorized int,o the product of the rot,ation matrix R. and the symmetrical matrix PO which can be diagonalized by an axis transformation as shown in equation (6) later. The physical significance of the above procedure corresponds to the transformation of the basis A into the new basis B in which the base vectors are the principal axes of the lattice deformation. i.e. %, (2:= 1 - 3) in Fig. 4, and to the deformation along the three orthonormal axes X,. Ross and Crocker’zo) and Acton and BevisQ*) have developed a theory of the martensite crystalIogr&phy involving the mechanisms of double shears on two independent shear systems. If two independent lattice invariant shears are admitt’ed to operate, hhe shape deformation P1 is expressed as

inhomogeneous shear planes, where 8 is the net angle of inhomogeneous shear. If two lattice invariant shears operate, the loci of.& and K, would produce the double shear cone and possible habit planes could be derived from the lines of the intersection of the initial cone of unextended line and the double shear cone.@@ The algebraic treatment for this case will be formally developed in the next se&ion. In 6he case of Cu-Zn alloy, the graphical met,hod involving manipulation on a stereographic net is, as shown in Fig. 5, almost the same as that in the est,ablished theories,(21*22)if the stereogram is to be viewed from t~he Hence, the cruystallographie features jz, direction. (4) p1 = &&PO&% of the mertensite with the 9R structure in Cu-Zn alloy could be formally derived in the similar way to where R, represents a rigid body rotation, end A1 that in the established theories,(22a23) if the rotation and A, represent the lattice invariant shears which of the axes is performed. tare denoted by 5, and S, in Ross and Cracker’s paper.(20t Since the m&rix Ai (i = 1, 9) has the similar form as H,, the magnitude of inhomogeneous 2.2 Algebraic treatments shear g,, the shear direction di and the shear plane Since the deformation specified by the process (i) p,’ are defined in the same manner as h,, 1, and mi’. is the distortion along three ort.honormal axes zi So far all the physical quantities such as lj, m,‘, P (i = 1 N 3), it is represented by the diagonal matrix C etc. have been referred to the basis A. However in C = (diag: A,, &, 4) (1) order to clarify physical significance of each matrix in equation (4), it would be appropriate to transform where A, and &, are the expansion along the x1 = the basis A into the basis B. When referred to the [ilO],, and x2 = fllO],l directions, respectively, and basis B, the matrices are tr&nsformed as follows & is the compression along the X, = [OO1]p, direction. Transformation of this matrix to the axes aligned along the cube edge directions of the parent phase (basis A) can be carried out by similarity transformation as ‘NT’, where T has colums Tij consisting of the components (X,), (i, j =1 N 3). The homogeneous shear direction 1, and plane m,’ yield the 8

F = U’P,U R = 7J’&RoU P = U’P*U Si = 0’A2TJ (i = 1, 2)

(5)

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where U has columns Uij consisting of the components (x*)> (i,j = 1 N 3) and P is diagonal. Thus, equation (4) can be rewritten as F = RPS,S,

where vu’ and k, are the values obtained on the basis B. 2.3 Application of the theory to the transformation in Cu-Zn alloy The lattice constant a, of the parent phase and those of the 9R structure, a, b and c of Cu-38.8 wt. % Zn alloy in terms of the orthorhombic coordinate are given in Table 1. The ratios, a:b:c deviate considerably from the ratios 3:1:3,/6 of the ideal close packed structureand moreover thee-axis makes an angle of 89’ with the a-axis. Therefore, strictly speaking, the 9R structure in Cu-38.8 wt. % Zn alloy is a monoclinic structure. The magnitude of the principal distortions are such that A1 = a/1/2a,, 1., = b/a0 and 1, = 1/2c/9a,,, as listed in Table 2, where the input data necessary for the calculation are also summarized. If the homogeneous shear and shuffling take place before the lattice deformat,ion, h, = 4 - tan 1” and h, = $, where tan 1” is the correction for the c-axis which makes an angle of 89” with the a-axis. Having known the L, (i = 1 -3) and hi (i = 1,2). we can describe the lattice deformation matrix B,, hence the principal axes and strains can be deduced from it. The lattice invariant shear of the 9R structure in Cu-Zn alloy is produced by the slip of partial dislocations with the Burgers vectors &(a/3)[1001pl.* which leave the stacking faults of the hexagonal type (+ sign) and cubic type (sign) behind. 1. Lettuce

martensite

constants of the parent a, b, e with 9R structure in Cu-38.8

a0 = 2.94 A,

a = 4.46 A,

b = 2.67

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TABLE 2. Input date necessary for calculation of the crystallography of mertensitic transformation

(6)

The physical significance of each matrix in equation (6) remains unchanged during the similarity transformation. Since equation (6) has the same form as that proposed by Ross and Crocker,(20) the crystallographic features such as the habit plane and the magnitude of the inhomogeneous shear can be derived from it in the same way as they did. However, since the vectors obtained with respect to the basis B, are of little practical use, the axes should be transformed again into the basis A. Then, the habit plane v’ and the direction of the shape deformation k are expressed as v’ = vu’U’ (7) k=Uk,

TABLE

VOL.

A,

phase a0 and wt.% Zn alloy

c = 19.3A

1.076348 l1 = -I2

,ig = 0.911263 = al = -a*

= [ I/\‘2

2s = 1.037735* --I/%‘2

ml’ = ml’ = pl’ = pz’ = (1/\‘2 hl = ) -

l/x’2

O] 0)

tan I”, h, = $

* Six decimal places are retained as purely device end do not reflect the accuracy of the 1,.

numerical

However, it is assumed in equation (6) or (4) that the slip occurs in an as yet untransformed parent phase. In this case, the slip direction are -&[ilO],l in accordance with -~-[loO]~,,, while the slip plane is (110),1 plane. If we choose the matrix A, and 4 so as to represent the slip in the direction [IiO]Dl and [IlO],,, respectively, the vectors di and p,’ have the values as shown in Table 2. In general, the magnitude of the inhomogeneous shear g2 must be assumed for the calculation of gl. However, in Cu-Zn alloy, since pl’ and d, are equal to p2’ and -d,, respectively, the product of A, and A, can be simply written by the following form A = Al& g =

= &A,

= I + gl,m,’ 6,

91 -

92.

Thus, we can make the calculation for the unknown quant,ity g without assuming any value for g2. Of the two values of g thus obtained. the smaller value will be chosen on the basis of the minimum strain energy, though the larger value gives a crystallographically equivalent solution. Only two possible habit planes or directions of the shape deformation exist for the smaller value of the shear magnitude g, but crystallographically these t,wo habit planes and directions of the shape deformation are also equivalent. The calculated values which represent the habit plane normal and the direction of shape deformation etc. are shown in Table 3 with the available experimental data of the martensitic transformation from the B2 to the 9R structure in Cu-38.8 wt. % Zn alloy, where the magnitudes of the inhomogeneous shear are expressed by the values of g, instelid of g, because the values of g1 and g2 are equal in magnitude but opposite in sign as verified later. Two different experimental data of g1 are cited in Table 3. The values shown in the upper row are those obtained by Kajiwara’26) presuming the existence of only one type of the l /?I’ denotes the martenslte produced from /&.

phase

with

9R

structure

KUBO TILE

AND

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MARTENSITIC

Habit plane normal (u’)

Direction (k) Shape deformation Magnitude (,a) Magnitude of shear (gr)

[00118, - COlOlP,’ (100)8, -

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3. Comparison of theoretical predictions with experimental results

Experiment

Orientation relationship

TRANSFORMATION

(104)8,’

Experimental 0iT0r

Theory

Disorepancy

Ref.

0.7191 -0.1271 0.6832

rt2”

0.7555 -0.1240 0.6433

2O59’

(25)

0.6915 -0.0905 -0.7166

f5”

0.6995 -0.1148 -0.7054

l”14’

(25) (25)

0.192

93”

0.2024

35’

(25)

0.0023

26’

44’ 4”21’

44’ 21’

(27) (25)

2’48’ 44’

2’48’ 44’

(15)

0.005 N 0.002 0.010

0 80

stacking fault i.e. cubic type, and the value in the lower row is due to the present work, taking it for granted that both the cubic and hexagonal type of the stacking faults are present. 3. DISCUSSION

3.1 Lattice invariant shear In order to facilitate the algebraic computation, the slip has been assumed to occur in an untransformed parent phase, and consequently the values of g are those referred to the basis A. Therefore, to 6nd out the relation between the magnitude of the lattice invariant shear gr and g2 and the stacking fault parameters a and /? defined by Nishiyama et aZ.“O) we consider the quasi-9R structure which will be formed from the parent phase by the processes (ii) plus (iii) without the process (i). Figure 6(a) shows sche-

matically the tilting of the [llo],, axis in this quasi-9R structure and Fig. 6(b) shows it in the faulted qu&-9R structure. In Fig. 6(a), the unit normal I&I of the (llo),, plane transforms to LM,, by the process (ii) and subsequently to LM’ by the process (iii). Since the vector LIE indicates the averaged inclination of the ( llO),L planes in the qmsi-9R structure, it can be divided into the vector LO which represents the total effect due to the cubic type stackings (c-type stackings) and the vector OM due to the hexagonal type stackings (h-type stackings). If the stacking faults are introduced into this qwtsi-9R structure, the c-type stackings denoted by LO change into those denoted by LP, PQ and QR as shown in Fig. 6(b), where LPrepresents the effect ofthe c-type stackings and both the vectors PQ and QR represent the effects of the #-type stackings. The vector PQ is derived taking into account only the inclination to the [IlO], direction of the h-type stackings continuing from the c-type stackings, and the vector QR indicates the effect due to the h’-type stackings continuing from the h’-type stackings. If the contributions of QR to the [llOIPl direction and to the [110],,,, direction are separated from each other, they can be denoted by QH and HR, respectively. Introduction of the stacking faults into the quuL9R structure also affects the vector OM’ in Fig. 6(a) and gives rise to the c-type stackings,represented by RS. Since g1 is the magnitude of the lattice invariant shear connected with the [lTO],, direction, it can be readily obtained by referring to Fig. 6 as g1= (Iti

FIQ. 6. Schematic representation of the quasi-9R structure. (a) The unit normal LH, representing (110)~~ plane in Cu-Zn alloy, inclines to LM’ by homogeneous shear LO and shuffling OhI’. (b) Change in slope of LO and OM’ by inserting stacking faults.

(26)

+ RS + HRI -

IUII) cos ‘p

(9)

Similarly qs =

(IPQ + SW + WI - IOaa’l) ~0s ‘p (10)

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Using the parameters a and @ defined by Nishiyama et uZ.os) we have (IW + RN cos (P == 3Wl

\ -

3 -

a2(l - B)i +. _ I)otr-3 20: - /9 r==3

[ER] COBp, = @HI cos rp =

3i%(l -

a)B

3-2201-B

}

(11)

ILO1cos Ip = 2 ]OM”] cos 9 = 2Nt (IP& +

SWl) cosy

it

@) a: Tz cr3

= 3Ntfl - a)‘(1 3-2a-@

where 3N is the number of the (1 1O),1planes per unit length of LM and t is the magnitude of the Burgers vectors f (u,/6) [l 10IP,. Substituting equation (11) in (9) and (lo), we obtain s1=

-32

e=lt--tr 1-B

I -e =

9pe

_

1 1)

(12)

i

It should be noted that gr is equal to g, in magnitude but opposite in sign. This means that if t,he stacking faults give rise to the new c-type stackings, the h-type stackings decrease as many as the newly produced c-types. Then we can readily estimate the value of gr from the obtained value of g, because g = 2gr. It is evident from equation (12) that if the c-type stacking faults are predominant (a > @), gr becomes positive, while if the h-type stacking faults are predominant gr becomes negative. In the ease of the 9R structure in Cu-Zn alloy, t,he calculated value of g, is positive and the c-type stacking faults

FIG. 7. Stacking fault prob8biiities of martensite with the 9R structure in Cu-Zn alloy. A straight line A B indicates the theoretical relation between CLand /3.

expected to exist predominamly. Actually, Kajiwara(a@ has estimated only the values of a from the electron diffraction patterns on the assumption that there is no stacking fault of h-type, because the W spots are too diffuse to find out their centers in the electron diffraction patterns. These values are plotted in Fig. 7 by the open circles in the a axis. We have also carried out the electron diffraction studies of the QR structure in Cu-38.8 wt. % Zn alloy. Although W spots are extremely diffuse in most diffraction patterns, it happened to be able to 6nd out the center of the W spot in Fig. 8(b), and the values of CLand /? could be determined to be 0,243 and 0.173 respectively, using the Kakinoki-Komura theory.o’j The result is shown in Fig. 7 by the solid circle with the hatched area to show the range of the experimental error. It should be noted that in this case also, c1is greater than 8. are

FIQ. 8. (8) !f!ransmission electron micrograph of marteneite with diamond figure in &x-38.? wt.% Zn i%lIoy. (b) Eleatron diffraotion pattern of martensite in (8) with the 9R structure.

KUBO

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The theory predicts a linear relationship between a and fi (cf. equation (12)). Physically it means that a and p are in the linear relationship, even though they may vary from crystal to crystal.* Substituting the theoretical value of g1 into equation (12), we obtain a straight line AB as shown in Fig. 7. It should be noted that the solid circle obtained in the present work lies near the line AB and the hatched area representing the range of the experimental error is also nearly parallel to the theoretical line. 3.2 Comparison of theoretical experimental results

values

with

The angle between the theoretical habit plane normal and the experimental one is 2”59’, while the limit of the experimental error is 2”. However, this discrepancy may be explained in the viewpoint of the accuracy of lattice constants. The lattice constants of the parent phase and the product phase (9R) must be known as accurately as possible to get reliable solutions for the habit plane normal and the magnitude of gl, etc. However, the lattice constants have been obtained only to an accuracy of the relative error of about 0.3 per cent, as shown in Table 1. Therefore, the theoretical values also contain errors of about the same amount. Thus, for example, if the dilatation parameter 6 = 0.997 which might be deduced from the accuracy of lattice constants is introduced into the theory, the habit plane normal is estimated as (0.7391 -0.1313 0.6606),1. In this case, the angle between the calculated habit plane normal and the experimental one is only l”44’ which is within the limit of experimental error. The direction of shape deformation is one of the most sensitive quantities to the variation of the lattice constants as well as the habit plane. The discrepancy between the theoretical and the experimental results has been found to be 1”14’, which is within the limit of the experimental error. As for the orientation relationship between the parent phase and product phase, two kinds of experimental data are available as shown in Table 3, which are obtained by X-ray and electron diffraction methods. The large discrepancy between the theoretical and experimental results obtained by electron * The stacking fault probabilities of PI’ (18R structure) martensite of Cu-12.3 wt.?& Al alloy have been obtained by analyzing the diffuseness and shifts of spots in the electron diffraotion patterns.‘1g1 The values of a and /? obtained from ten different martensites are in the range of 0.004-0.297 and 0.115-0.412 respectively, varying from crystal to crystal, but they fall in the straight line of B = (0.813 f 0.127)a + (0.151 f 0.024), using the least square method.

TRANSFORMATION

16’75

diffraction method and shown in the lower row may partly be attributed to the experimental error, because the electron beam which deviates in several degrees from the exact Bragg angle can also give rise to the diffraction patterns. 4. CONCLUSION

The crystallography of the martensitic transformation with long period stacking order has been developed and applied to the transformation from the B2 to 9R structure in Cu-Zn alloy. The point in the theory is to represent the matrix B, of the lattice deformation by the product of three matrices which represent pure deformation, homogeneous shear and shuffling. It is revealed that if the principal axes of the lattice deformation are chosen as the reference axes, the theory could be treated in the same way as the established theories. Combining the Kakinoki-Komura theory of the stacking fault probability with the present theory, it is shown that (1 - a)/(1 - /?) is constant, where a and /I represent the stacking fault probabilities of the cubic type and hexagonal type, respectively. In order to confirm this relation, electron diffraction studies have been carried out on the 9R structure of Cu-Zn alloy. Application of the theory to the martensitio transformation in Cu-Zn alloy is found to give results in satisfactory agreement with the available experimental data. REFERENCES 1. H. SATO, R. S. TOTE and G. HONJO, J. Phy8. Chem. Solids 28. 137 (19671. 2. H. SATO,‘&.S.‘?OTH, G. SEIRANE and D. E. Cox, J. Phys. Chem. Solids 27, 413 (1966). 3. H. SATO,R. S. TOTE and G. HONJO, Solid State Commun. 4, 103 (1966). 4. K. BURKEARDT and K. SCHUBERT,2. Metallk. 58, 864 (1965). 5. R. S. TOTE and H. SATO, Appl. Phys. Lett. 9, 101 (1966). 6. J. WERT and K. SCHUBERT,2. MetaUk. 49, 533 (1958). 7. B. C. GIESSENand N. J. GRANT, Acta Crystallogr. 18, 1080 (1965). 8. Z. NISHIYAMAand S. KAJIWARA, Jap. J. Appl. Phy8. 2,

475 (1963).

9. P. R. SWAN and H. WARLIMONT,Acta Met. 11,611 (1963). 10. M. WILKENS and H. WBRLIMONT, Acta Met. 11, 1099 (1963); H. WARLIMONT and M. WILKENS, 2. MetaUk. 65, 382 (1964); H. WARLIMONT and M. WILKENS, 2. MetaUk. 56, 850 (1965). 11. H. SATO, R. S. TOTE and G. HONJO, Acta Met. 15, 1381 (1967). 12. R. S. TOTE and H. SATO, Acta Met. 15,1397 (1967). 13. R. S. TOTE and H. SATO, Acta Met. 16, 413 (1968). 14. M. HIRABAYABHI,N. INO and K. HIRA~A, J. Phye. SOC. Japan 29, 1509 (1967). 15. S. SATOand K. TAKEZAWA, Proc. Int. Conf. on Strength of Metala and AZZoy8, Tokyo, (1967); Trans. Japan Inat.

Met. 9, SuppZ. 926 (1967). J. Phys. Sot. Japan SO, 1757 (1971).

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17. J. KAKINOKI end Y. KOMIJBA, Acta Cryutallogr. 19, 137 (1965). 18. J. KAKINOKI, Acta Crystallogr. 23, 876 (1967). 19. Z. NISHIYABLA,J. KAEINOKI and S. KAJIWARA, J. Phye. Sot. Japan 20, 1192 (1965). 20. N. D. H. Ross and A. G. CROCKER, Acta Met. 18, 405 (1970). 21. D. S. LIEBERIUN, Acta Met. 6, 680 (1958). 22. J. S.Bow~~sand J. K. MACKENZIE, Acta Met. 2,129,224

VOL.

23. 24. 25. 26. 27.

21, 1973

(1964); J. K, MACKENZIE and J. S. Bowms, Acta filet. 2. 138 (1964); 5, 137 (1957). M. S. WECHSLER, D. S. LIEBERMAN and T. A. READ, Tram. Am. Inst. Min. Engra. 197, 1503 (1953). A. F. ACTON end M. BEVIS, Mater. Sci. Eng. 5, 19 (1969). R. D. GUWOOD and D. HVLL, Acta Met. 6, 98 (1958). S. KAJIWARA, J. Phys. Sot. Japan SO, 768 (1971). W. JoLLY~~~D.HuLL, J. In&. Metals 92,129 (1963-64).