The lattice spacings and the stability of close-packed hexagonal Cu-Ga, Cu-Ge and Ag-Al alloys

THE

LATTICE SPACINGS AND THE STABILITY HEXAGONAL Cu-Ga, Cu-Ge AND A&Al T. B. MASSALSKIt:

AND

OF CLOSE-PACKED ALLOYS*

B. COCKAYNET

The lattice spacings of close-packed hexagonal i-phases in the systems Cu-Ge, Cu-Ga and Ag-Al have been determined as a function of composition and electron : atom ratio, using alloys quenched from 550%. The results how that overlap of occupied states across the { lOi0) faces of the Brillouin zone occurs at the electron : atom ratio of 1.415 in Cu-Ge alloys, 1.43 in Cu-Ga alloys and at 1.47 in Ag-Al alloys. In all three systems the relative distortion of the a lattice spacings due to overlap increases linearly at higher values of overlap at a rate of 3.5 x 10-a A per 0.1 increase in the electron : atom ratio. In the system Ag-Al the change of slope in the c lattice spacings near the electron : atom value of 1.62 may be interpreted in terms of interaction of the Fermi surface with the (1010) faces of the Brillouin zone before contact. LES

DIMENSIONS DU RESEAU COMPACTE DANS

ET LES

LA STABILITE DE LA PHASE HEXAGONALE ALLIAGES Cu-Ga, Cu-Ge et Ag-Al

Les auteurs ont trempe It partir de 550°C des alliages de Cu-Ga, Cu-Ge et Ag-Al et determine les dimensions du reseau de la phase 5 hexagonale compacte en for&ion de la composition et du rapport electrons : atomes. Les resultats revelent qu’il y a un chevauchement des &tats occupes sur les plans {lOiO} des zones de Brillouin pour les rapports electrons : atomes de 1,415 pour les alliages de Cu-Ge, de 1,43 pour ceuse de Cu-Ga et de 1,7 pour les alliages de Ag-Al. Ce chevauchement entraine dans les trois systemes une distorsion relative du parametre “a” du reseau qui augmente d’une fapon liniraire a raison de 3,5 x 1O-3 A pour un accroissement de 0,l des rapports electrons : atomes. Dans le systeme Ag-Al, le changement de la pente du parametre c du reseau aux environs d’un rapport, electrons : atomes de I,62 peut etre interpret& comme une interaction des surfaces de Fermi cvec les plans {lOiO} des zones de Brillouin avant contact. GITTERKONSTANTEN

UND STABILITAT DER HEXAGONAL LEGIERUNGEN Cu-Ga, Cu-Ge und Ag-Al

DICHTGEPACKTEN

Die Gitterkonstanten der Z;-Phasen (hexagonale dichteste Kugelpackung) in den Systemen Cu-Ga, Cu-Ge und Ag-Al wurden als Funktion der Zusammensetzung und des Verhaltnisses Elektronen : Atome bestimmt. Dazu wurden die Legierungen von 550” C abgeschreckt. Das Verhaltnis Elektronen : Atome, bei dem die beset&en Zustande die {lOTO}-Flachen der Brillouin-Zone iiberlappen, betriigt 1,415 fur Cu-Ge-, I,43 fur Cu-Ga-Legierungen, dagegen 1,47 fur Ag-Al-Legierungen. In allen drei Systemen steigt die relative dnderung der Gitterkonstanten a infolge der Uberlappung linar, und zwar urn 3,5 x 10-3, wenn das Verhaltnis Elektronen : Atome urn 0,l steight. Beim System Ag-A1 andert sich beim Verhaltnis Elektronen : Atome 1,62 die Neigu der Kurven der Fermi-Oberfur die Gitterkonstante c. Das lasst sich verstehen, wenn man eine Wechselwirkung fliiche und der {lOil}-Fliichen de Brillouin-Zone bereits vor ihrer Bertihrung annimmt.

1. INTRODUCTION

Intermediate gonal

structure

(c.p.

hex.)

systems which copper, elements When

are often

hexa-

stable

in the

of the periodic

are present

the ranges

scatter

when

expressed

values of e/a, the most frequently

in

terms

of

observed values are

in the region either of 312 or 714. Nevertheless

silver and gold form with the

of the B-subgroups

such phases

homogeneity

considerable

phases with the close-packed

almost

all such phases can be classified as electron compounds,

table.

and they are believed to be stabilized

of their

by interaction

of the Fermi surface and certain faces of the Brillouin

lie within the broad region of electron

zone.(3*4)

Information

about

the

nature

of

such

concentration values between approximately 1.3 and 1.8, and they are often referred to as electron com-

interaction can usually be obtained from the study of the variation of lattice spacings with composition, and

pounds.(1-3)

e/a. All the c.p. hex. phases so far examined

The term

electron

concentration

usually denotes the ratio of all valency the number of atoms.

(e/a)

electrons

to

remarkably

are overlapped, the a lattice spacings, extending in the direction at right angles to the {IOiO} faces, expand, and the axial ratio decreases;(2*3) if the interaction takes place between the Fermi surface and the {OOOZ}

* Received December 12, 1958. t Dept. of Physical Metallurgy, University of Birmingham, Edgbaston, Birmingham 15, England. 2 Now at Mellon Institute Pittsburgh 13, Pa., U.S.A. METALLURGICA,

VOL.

7, DECEMBER

1959

show

trends.

If, as a result of the increase in the electron concentration, the {lOiO} faces of the Brillouin zone are about to be touched by the Fermi surface, or if they

Previous investigationsP3) have shown that although the positions, and widths, of individual phases in different equilibrium diagrams show quite

ACTA

consistent

faces of the zone the c lattice spacings expand and the 762

MASSALSKI

axial ratio increases. ratio

distortions

Goodenough’@,

AND

COCKAYNE:

The mechanisms

have

been

Schubert(‘)

LATTICE

for such axial

discussed

by

Jonesc5),

changes of the lattice spacings and of the

axial ratio observed

in c.p. hex. electron compounds.

If the composition

of alloys

in a ternary

adjusted so that no change of e/a occurs

system

is

the axial ratio

remains strikingly constant.t4) The present investigation is an extension

of lattice

spacing measurements

already made on other systems

based

silver

upon

copper,

and

gold.

lattice spacing have been accurately systems

copper-gallium,

silver-aluminium, 1.8.

The

provides

AND

The approximate

763

STABILITY

temperature

at which measurements

were recorded was 26” & 2°C.

and others.

Previous investigations have also shown that the e/a is the major controlling factor which influences the systematic

SPACINGS

Changes

at values of e/a between of

new information

the

3.1

data

so

RESULTS

The system silver-aluminium

The values of axial ratio and a and c lattice spacings obtained

for the c.p. hex. alloys quenched from 550°C

are plotted spacing

as a function

varies linearly

the axial ratio

of e/a in Fig.

1.

and the c spacings

SILVER

show a distinct

ALUMINUM

I6300

4 6600

and 1.36 and

- 4.6500 .g

obtained

B

about the onset, and influ-

- 4 6400

.?

ence, of overlap across the {lOiO} faces of the Brillouin

2

zone in all three systems, and the nature of interactions between the approacmng

Alloys

were

conductivity standardized miniurn. melted

AND

EXPERIMENTAL

prepared

from

and

from

this

gallium,

germanium,

induction

high

silver

and

alu-

Weighed quantities of the pure metals were under

reduced

technique

Alloys containing crucibles,

METHODS

spectroscopically

pressure

of argon

sealed silica capsules with vigorous of

I5900

oxygen-free,

copper,*

4.6300

\

Fermi surface and the {lOil}

zone facrs in the system silver-aluminium. 2. MATERIALS

have

been

aluminium

shaking;

published

in small, details 1.50

earlier.(*)

were melted in graphite

under argon, by means of a high-frequency Details of homogenization, technique.

FIG.

155

160 Electron

change of slope between

the same as in earlier investigations.(s,s)

decrease

In order to obtain

accurate

values

of the lattice

1.63.

1.65 1.70 Concentration

has already

been

for the axial shown

the precision .necessary for revealing now observed.

against

the Nelson

and

Riley functionoO) and the a spacings were then evaluated by extrapolation to 8 = 90”. To obtain a straight-line

plot

the axial ratio was first adjusted

by a trial and error method.

The c lattice spacings

were then calculated from the evaluated a and c/a All values are reported in Angstrom units, values. based upon the following CUK, radiation:

values

assumed

CuK,,

= 1.5405006 A

CuK,,

= 1.5443412 A

for the

Supplied by Imperial Chemical Industries Limited. t Supplied by Messrs. Johnson, Matthey and Co., Ltd.

ratio

to

by the work

of

Westgren and Bradley in 1928ol) but their data lacks

into

were plotted

I80

the e/a values of 1.62 and

The general tendency

spacings the last eight high-angle X-ray lines, resolved doublets,

1.75

of lattice spacings with electron 1. Variation concentration in the system silver-aluminium.

heat-treatment, metallographic examination, chemical analysis, and preparation of X-ray specimens were

*

The a

with e/a, but the curves for

in

measured in the

copper-germanium,

interpretation

3. EXPERIMENTAL

3.2

the finer details

The system copper-germanium

The present lattice spacing values for the c.p. hex. alloys

quenched

from

550°C

are shown

in Fig.

2,

plotted again as a function of e/a. It can be seen from the figure that the c spacing curve shows a general decrease with increasing e/u. The curves for axial ratio, and a spacing, however, show a change of slope as the electron to atom ratio increases from 1.36 to 1.45. At values of electron concentration greater than 1.45, axial ratio and a spacing increase linearly as e/a is increased.

764

ACTA COPPER

t

METALLURGICA,

GERMANIUM

VOL.

7,

195!)

present results for two alloys in the ii phase, quenched

I

from 550°C a,nd one alloy in the 5, phase, quenched from 450°C are listed in Table

2.6050

1 together

alloy in the CI.phase, quenched

i

with one

from 550°C.

These

results are plotted in the lower portion of Fig. 3. TABLE

1.

---

/

1.6350

2.5950

2

2 2

i .o 1.6300

/

Compositioni I ! I c.p. of alloy n f&C. a e.p. chex. Cu(&%) ‘Phase (A) *Iex. / j (& j (A) Ga

:

c/a

2.5900 o

‘i; c

I

80.7 78.5 78.0 77.8

z 4” 1.6350 I

e/u

i

19.3 21.5 22.0 22.2

,

K

/

g2

1

3.6991 ‘1;2.,946)! / 2:5992 / 4.2442 -. 2 6007 ’ 4.2433 i 2.6013 4.2435

<: ! _ _.

’

-

/ 1.6329 1.6316 1.6313

i

1.386 1.430 1.440 1.444

I~

_1... _~ -‘_- ~_._ ..- _

~-

‘k200L0 Electron

Concehtruti~a?

Variation lattice spacings with electron centration in the system copper-germanium.

FIG.

2.

The present results differ consid~rabIy reported by Schubert and Brandauer(12) are

however

insufficiently

critical

eon-

the c.p. hex. phases are slightly

from those whose data

for

a

direct

comparison. 3.3

The result for the primary solid solution (M) agrees well with previous work,(isJ4) but the m spacings of previously This

reported

discrepancy

different

were measured;

The previous

accurate work on the c-p. hex. phases

in this system was done by Raynor

and Massalski(*)

higher than those

Raynor

is however

temperatures

a temperature

The system copper-g&km

by

and

~ssalsk~(4).

consistent,

with

at which the lattice

the

spacings-

the present work was carried out at of 26’ + 2”C, whilst the earlier work

was at 18” & 3°C.

4. DISCUSSION

spacings

of the & and

4.1

5s phases shown in the upper portion

of Fig. 3. The

faces qf the Rrillouin zone

GALLIUM

observed in all c.p. hex. phases of systems based upon

who determined

the lattice

Overlap of the Fermi

The COPPER

present

results

across the {lOiO:o)

surjbce

confirm

the

trend

already

C!u, Ag and Au for the axial ratio to dezease

with

increase in e/a if the value of e/a is approximately between 1.36 and 1.85. comparisons

In addition certain interesting

can be made between the three systems

stuclied if the a spacings available for the primary solid solutions

and t*he u spacings obtained

in the present

work for the c.p, hex. phases are plott,ed as functions of composition face-centred 2.610 2.600

O-Present Work l - Previous Work

hexagonal

respectively,

& 2.590

structures

;; “, 2.570

a/2/2

zi 5 2.580

planes.*

135 140 h45 wec,ronConceowDIIo” Atomic Per Cent Gallium

FIG. and

structure

and

3. Variation of lattice spacings with composition electron concentration in the system coppergallium.

the

close-packed

structure possess the same arrangement

atoms in their closest packed planes,

-

2.560

as shown in Figs. 3, 4 and 5. Since the cubic

it is convenient

(11l> and

to compare

of

(OOOI>

t‘he two

by referring to the lattice spa,eings in these It may be seen from Figs. 3 and

4 that the

spacing of the primary solid solutions of copper

with gallium, and copper with germanium, increases as alloying element is added. This is in contrast to the primary

solid solution

of silver with al~lnlinium, see

* For tht purpose the following correspondence is valid: spacing along the cube edge = -\/2 x spacing in the cubic 1111) pianos =spacing in t,be c.p. hex. {OOOl} basal planes. Therefore cubic a spacings must be divided by .~‘2 for direct comparison with the c.p. hex (I spacings.

MASSALSKI

AND

COCKAYNE:

LATTICE

2.600

J

f 2.590 ? a” 2.580 c) 2.570

o-Present l -Previous

2.560 2.550

FIG. 4.

0

765

i -_i

Variation of lattice spacings with composition in the system copper-germanium.

Fig 5, where the a/2/2 spacing decreases with increasing aluminium content According to Raynor such lattice spacing changes may be interpreted in terms of the size and valencies of the component atoms. It appears from Fig. 3, that these size and valency effects continue with the hexagonal &- phase of the copper-gallium system, but that in the J&phase the

2.590 2.580 2.570 2.560 2.550

800

SILVER

ALUMINIUM

Electron Concen:rotion FIG. 6. Variation of lattice spacings with elentron concentration in the systems copper---germanium and An shows the lattioe distortion due silver--aluminium . to overlap.

p 700$SOO-

P

g.500

J-l.

E

c” 400

STABILITY

4

Work Work

I IO 20 Atomic Per Cent Germanium

AND

a sFacing/composition curve has an increased slope. * The size and valency effects also continue in the c.p. hex. phase of the copter-germanium system up to about 14 at. Od,germanium; at greater germanium contents the a spacing curve increases in slope. In the silver-aluminium system, however, there is no change of slope within the c.p. hex. phase, but a marked change of slope, with a reversal of trend, exists between the curves for the primary solid solution and the c.p. hex. phase, as is evident from Fig. 5. If the lattice spacing changes arc plotted in terms of e/a it becomes apparent that a change of slope in the

2.610 %

SPACINGS

i

3001!--

/lLw

/II\

I

f

\

Temper&we

I

\

--.-.I--

4

‘2.890 -Present

Work

‘; 2.880 f E 2 ao 2.870 c) ,I I I I & J 2.8SOi 20 0 40 60 80 100 Atomic Per Cent Aluminium FIG. 5. Variation of lattice spacings with composition in the system silver-aluminium.

spacing occurs in all three systems at approximately the same value of electron concentration. This is illustrated in Fig. 6 for the systems copper-germanium and silver-aluminium. In the former case the change of slope occurs within the 5 phase, and in the latter case it falls intro the two-phase region between the a and 5 phases. Similarly, in the system coppergallium the change of slope occurs in the e/a region between the & an5 & phases. This is shown in Fig. 3. It is suggested that the observed change of slope is the result of a,n overlap of the Fermi surface across

n

* In order to confirm this point great care was taken to obtain high accuracy in lattice spacing measurements. The width of the phase fields {I and & is very narrow and only one alloy was studied in the c2 range, and two alloys in tbe [I range. Nevertheless we believe that the accuracy of the present measurements oateblishss the change of slope beyond any doubt.

ACTA

700

the {lOiO}

faces of the Brillouin

According

to Jonests),

overlaps

METALLURGICA,

VOL.

7,

1959

zone (see Fig. 7). tend to distort

Brillouin zone so that the overlapped

the

zone face moves

towards the origin of the K-space, thus causing an expansion of the corresponding lattice spacing in real space.

It has been proposed

earlierc4) that all

c.p. hex. 312 and 714 electron compounds by an overlap

across

are stabilized

the {lOiO> faces of the zone

which therefore causes the a spacings to increase and the axial ratio to decrease. that such an overlap in the vicinity

The present work suggests

actually

begins at values of e/a

of 1.42 in copper based alloys and 1.47

in silver based alloys, and is superimposed continued

decrease,

or increase,

in the system

in the systems

upon the

silver-aluminium,

copper-germanium

I.4

and

copper-gallium of the a lattice spacing curve, which would still take place had there been no overlap. For

convenience,

the

distortion

due

to

overlap

within the c.p. hex. phases at any particular value of e/a may be defined as Aa, the difference between the observed

a lattice

spacings

by extrapolation solution.

This is indicated

fractional distortions value) are plotted electron straight due

systems,

in Fig.

6.

solid

If now the

ha/a (where a is the extrapolated for each system

as a function

of

concentration, as shown in Fig. 8, a set of lines is obtained showing that distortions

to overlap

identical.

and that value obtained

of the curves for the primary

are nearly

the same

in the three

and that the rates of distortion The extrapolation

are nearly

of the Aa/a vs. e/a plot

to the point where Aala is equal to zero, see Fig. 8, permits the establishment actually

begins.

of the point where overlap

The values obtained

are 1.43, 1.415

I.5 Electron

1.6 1.7 Concentration

I

FIG. 8. Variation of the relative distortion of the a lattice spacings (A+) with electron concentration in the systems copper-gallium, copper-germanium and silveraluminium.

and 1.47 respectively for the three systems copperand silver-aluminium. gallium, copper-germanium Since these values are very nearly the same in the binary alloys the method measurements

could

of accurate

be fruitfully

study of ternary c.p. hex. electron

lattice spacing

extended

to the

compounds.

For

example one could determine the valency contribution from a transition element by assuming that the points where Aala = 0 in a ternary system based on copper and silver as solvents correspond with the values of e/a equal to 1.42 for copper based alloys, and 1.47 for silver

based

alloys.

Work

of this kind

is now

in

progress in the authors’ laboratory. In the system copper-gallium the point of overlap falls between the two c.p. hex. phases 5s and &. The greatly restricted homogeneity have

been

previously

ranges of these phases

discussed

by

Raynor

and

Massalski(4) who proposed that they are a result of a borderline size factor* of gallium with respect to copper.

The present work now suggests that the [i

phase differs from the
with composition.

germanium, overlap

In the system

where the size-factor

is accommodated

copper-

is favourable(4)

the

within the wide c.p. hex.

phase, but in the system copper-gallium

more critical

size conditions result in the existence of a two-phase field between the 5s and [i phases at temperatures near 475°C. This is in close analogy to the two-phase field existing between the E and q phases in the system PIG. 7. Vertical section through the first Brillouin zone for the close packed hexagonal structure, proposed by Jones’s’. The contours inside the zone outline the most likely distortion of the Fermi surface at various values of the electron concentration.

d solute - d solute x 100 * Size factor may be defined as ~~-~d solute where d values denote the closest distance of approach in the structure of the pure element.

MASSALSKI

COCKAYNE:

AND

LATTICE

SPACINGS

Cu-Zn.(5) In the former case the 5s and & phases are separated by the {lOiO> overlap and in the latt,er case

approximately

the

mation

E and 7 phases

are separated

by

the

(0002}

overlap. 4.2

Wittig(ls)

AND

STABILITY

767

1.62, are the areas marked y in Fig. 7.

has recently

determined

the heat of for-

of c.p. hex. silver-aluminium

alloys and his

data indicate a minimum in the heat of formationjcom-

The interaction between the Fermi surface and the

{ lOi1) faces of the Brillouin zone

position

curve,

which

coincides

with the change

work.

Following

the minimum there is a steep rise of

The change of slope which occurs in the axial ratio and the e spacing curves of the silver-aluminium

the values

of the heats of formation,

c.p. hex. alloys, as shown in Fig. 1, can be related to

decreasing

stability

thermodynamic

the interaction between

effect,

the Fermi

the Brillouin zone.

proposed

by Goodenough@),

surface and the (lOil} faces of Such an interaction should tend to

move the {lOiO} faces towards the origin of the zone. For a c.p. hex. structure with an ideal axial ratio, the interaction

should occur at an electron concentration

of 1.67 provided

the Fermi

surface

is assumed

a

If the

data are correlated with the presently

the rise in the values of the heats of formation e/a increases above of the relatively within the zone.

1.62 is associated unfavourable

when

with the filling

states

(marked

y)

CONCLUSIONS

The experimental

results presented

above show that it is possible

The present work shows that in the system silver-

indicating

of the c.p. hex. phase.

discussed picture of the Brillouin zone it appears that

to

possess a spherical shape.

of

slope in the c lattice spacings observed in the present

aluminium

a change of slope occurs in the c and c/a

measurements

curves

values

onset of overlap across the {lOlO) faces of the Brillouin

at

However,

of

the axial

e/a ratio

between

1.61

in the region

and

1.64.

where the

interaction occurs is nearly 1.61, which is substantially below the ideal value of 1.63. Consequently, the zone will be extended in the direction of the (0002) faces and

of the lattice

and discussed

by means of accurate spacings

to detect

zone in the c.p. hex. electron compounds copper and silver.

the

based upon

The value of e/a at which overlap

occurs in the three systems studied varies from 1.415 to 1.47, and the relative rate of lattice spacing

such an extension presumably affects the (lOil} faces in such a way that the Fermi surface approaches them at values of e/a somewhat lower than 1.67. In addition

distortion due to overlap appears at higher values of overlap to be almost identical in the t,hree systems,

it is reasonable

3.5 x lop3 in the a lattice

to expect that the Fermi surface will

be considerably because

distorted

from

the spherical

shape

various indicated

to a relative

contours for the Fermi surface, for values of e/a, are shown in Fig. 7.

change

of approximately

spacing

per 0.1 change

in e/a. It is suggested

of the nearness of the zone discontinuities.

The most probable

amounting

that the presence

of two

closely

related c.p. hex. 5s and & phases in the system coppergallium may be a result of a combination

of borderline

Fig. 7 also shows that because of the presence of the

size effects discussed

{lOiO}

discontinuities

within the 5s phase discussed in the present work.

Fermi

surface will most likely interact

portion point

of each {lOil} marked

spacings range

expected

X.

zone, the

only with a

face, in the vicinity

The

presently

obtained

Evidence

which

suggests

that

the

observed change of slope in the c lattice spacing/com-

lattice

position curve of silver-aluminium

alloys is the result

of an interaction

between

the

change of slope in the c lattice spacings in

portions of {IOil} are touched.

Brillouin

zone faces, before they

the of

e/a

silver-aluminium values

where

alloys an

show

interaction

is

Fermi surface and > faces of the zone. Such a change

Professor

the (lOi1) faces of the Brillouin zone were to undergo a twist with the accompanying displacement of the

stimulating

increase while the a spacings unaffected.

may remain relatively

After the upper portions of the (lOTi} faces are touched the remaining portions of the Brillouin zone be filled,

as e/a increases

above

the values

of

surface

This work was done under the general direction

of slope would be expected if, as a result of interaction,

{OOOZ} faces towards the origin of the zone. As a result of such a distortion the c spacings should show an

Fermi

and

ACKNOWLEDGMENTS

between the approaching

a portion of the (lOi

to

is discussed

of the a

in

significant the

within the Brillouin

earlierc4) and the overlap effect

G. V. Raynor

the provision

of facilities

of

to whom we are grateful for for research and for many

discussions.

thank the University

One of us (B. C.) wishes to of Birmingham for the provision

of financial support in the form of the William Gibbons Research Scholarship. REFERENCES 1. W. HUME-ROTHERY and G. V. RAYNOR, The structureof metals and alloys. Institute of Metals (Monograph and Report series No. 1) London (1954).

768

ACTA

BIETALLURGICA,

2. T. B. MASSALSKI, 1955 Sywqvosium on the theory of alloy phases. Amer. Sm. Metals, Cleveland (1956). 3. T. B. MASSALSKI,The lattice spacing relationships in alloys Metallurgical Reviews 3, 45 (1958). 4. G. V. RAYNOR and T. B. MASSALSKI,Acta Met. 3, 480 (1955). 5. H. JONES, Proc. Roy. Sot. A147, 396 (1934); Phil. Msg. 41, 663 (1950). J. GOODENOUQH,Phys. Rev. 89, 282 (1953). 43, 1 (1952). t : K. SCHUBERT,Z. 8. T. B. MASSALSKIand 0. V. RAYNOR, J. Inst. Met. 82, 539 (1954). 9. T. B. MASSALSKI,Acta Met. 5, 541 (1957). 10. J. B. NELSON and D. P. RILEY, Proc. Phys. Sot. A57, 160 (1945).

VOL.

7.

1959

11. A. WESTGREN and A. J. BRADLEY, Phil. Mag. 6, 280 (1928). 12. K. SCHUBERT and G. BRANDAVER, 2. Metallk. 43,262 (1952).

13. E. A. OWEN and E. W. ROBERTS, Phil. Mag. 27, 294 (1939). 14. R. J. HODGKINSON, Phil. Msg. 46, 410 (1955). 15. W. HUME-ROTHERY, G. F. LEWIS and P. W. REYNOLDS, PTOC. Roy. Sot. A157, 167 (1936). 16. F. FOOTE and E. R. JETTE, Trans. Amer. Inst. Min. (Metall.) Engrs. 143, 151 (1941). 17. G. V. RAYNOR, Trans. FmxZay Sot. 45, 698 (1949). Symposium 01~ Physical Chemistry 18. F. E. WITTIG, N.P.L. of

Metallic

Solutions

and

Intermetallic

London (1958) to be published.

Compounds.