Stability and axial ratios of various close-packed structures

STABILITY

AND AXIAL

RATIOS

OF VARIOUS

CLOSE-PACKED

STRUCTURES*

C. H. HODGES? The structural energy of various close-packed phases has been analysed using the method of the interactions between pairs of close-packed planes first formulated by Blandin, Friedel and Saada .‘I) Using this method we explain a correlation between the structure and the deviation of its axial-ratio from ideality, which exists in the rare-earths and in one or two other cases. 1Ve also discuss the conditions for the appearance of complex phases such as the d.h.c.1~. and Sm structures. STABILITE

ET

RAPPORTS

AXIAUS

DE

DIVERSES

STRUCTURES

COMPACTES

On analyse 1’Bnergie de st,ructure de diverses phases compactes en utilisant la m&hode des interactions ent,re paires de plans compacts form&e pour la premii?re fois par Blandin, Friedel et Saada.‘ll En utilisant cette m&hode, on explique une correlation entre la structure et 1’6cart B l’idbalitb de son rapport axial, qui existe dans les terres rares et, dans un ou deux aut,res cas. On discute aussi les conditions de l’apparition de phases complexes telles que les strorturcxs h.c. double et Sm. STABILITKT

TJND

ACHSIALT’ERHALTNIS

MEHRERER

DlCHTEST-GEPACKTER

STRUKTURES Methode der Wechselwirkung Mit Hilfe des zuerst van Blandin, Friedel und Saada (‘) formulierten zwischen Paaren dichtest gepackter Ebenen aurde die strukturelle Energie verschiedener dichtest grpackter Phasen analysiert. Es wird auf diese Weise eine Beziehung zwischen der Struktur und der Abweichung ihres Achsialverhiiltnisses vom Idealwert erkliirt, wie sie in den seltenen Erden sowie in ein oder zwei anderen Fallen existiert,. Frrner wcrden die Baclingungen fiir das Auftreten komplexer Phesen wie d.h.c.p. und Sm diskutiert.

INTRODUCTION

methods

Recently, calculation

have

of stacking-fault

fundamental

point

pseudopotentials

of

and

their work, Blandin, a method of

the

whereby

only

the

based

on

the

the problem between

of

of stacking

faults,

structure

The f.c.c.

and h.c.p.

(Sm, AB

2). not

of “close-

in Mg metal,

Ag-In

and Ag-Sb

pressure(3) and in the Au-In(*) systems at normal pressure. is observed

in

various

not only in the under

high

and Au-Gac5)

alloy

alloy

The Samarium structure

rare-earth

Samarium,

when these are subjected

andinintra

rare-earth alloysystems.(7)

various stacking sequences.

has postulated

By considering

the energies of formation fault, Blandin,

derive limits of stability with electron

per atom

Friedel

of various

the rare-earth

and Saada

of the h.c.p. and f.c.c. phases ratio e/a.

very

but also more recently

i.e. those structures crystal structures, packed” formed by stacking close-packed planes of atoms in

types of stacking

ABCBC

phases are, of course,

rare earths and their alloys,

but also the

and stability

Lanthanum

,4 C).

has now been quite widely reported

close-packed

A of Ref.

close-packed,

,4 B A C A B A C)

common in metal and alloy systems. The d.h.c.p. phase

is analysed in terms

pairs

(d) the Samarium

In

originated

hexagonal

sbructure (d.h.c.p.:

of

theory.(lsa)

in Appendix

of the occurrence

t’he

use

is a very useful one in discussing

energies

question

for

energies in metals from a

view

perturbation

(also described

This method

evolved

Friedel and Saada’l)

interactions

planes

(c) double

been

with

It is clear from

periodic

besides

to pressuret6) Jayaraman(6*8)

that there is a general trend among metals h.c.p. +

increasing

that occurs

metals,

density.

Sm --f d.h.c.p. + f.c.c.

This

is the

same

as we cross the rare-earth

trend

row in the

table from right to left as far as Ce leaving

their work that for some e/a ratios both the f.c.c. and

out Eu, Yb which are divalent instead of trivalentPI,

1l.c.p. phases

to faulting

or as we alloy

that other

concentrations

(see Table close

packed

stacking

are unstable

6 of Ref. phases

sequences

with respect

1) and this means

having

more

should become

h.c.p. or f.c.c. in these regions.

complicated

more stable than

The phases we shall

consider in this paper arc : (a) face-centred cubic (f.c.c., A B C A B C) (b) hexagonal-close-packed

* Received April 19, 1967. t Cavendish Laboratory, Cambridge. Now at: Laboratory of Atomic and Solid State Physics, Cornell University. ACTA

METALLCRGICA,

VOL.

15, DECEMBER

1967

rare-earth

with increasing

We shall use the method

of interactions

planes to explain a striking correlation bimes exists

between

from ideality

and the particular

We

(h.c.p., A B A B A B)

a heavy

of a light rare-earth.(7)

shall also discuss

z In quoting the pendicular distance Thus t,o obtain c the must be divided by cell.by4.5. By ideal of ng:ld spheres. 1787

the the

deviation

in axial

structure

conditions

between

which someratio

adopted.$

under

which

c/a or axial ratio, c will refer to the perbetween next nearest neighbour planes. length of the true d.h.c.p. hexagonal cell 2 and that of the true Sm rhombohedral we mean c/a = 1.633 as for tho stacking

1788

one

ACTA

would

expect

Sm as complex

the

appearance

intermediate

METALLURGICA,

of d.h.c.p.

and

of crystal

structures the

structure In recent metals gaps

in

considering electrons,

changes

that in many

of the band

structure

gas are small.oO)

by the theory

gaps and matrix

potential

metals

from

This

of pseudopoten-

elements

first

method

was

Harrison,(15)

Sham(16) theory

pseudopotential.

It

scheme

formulated

and up

by

who

second

has recently

Heine and Weaireos)

in discussing

ratio in the divalent

metals.

be

order

been

the

applied

by

the trends in c/a

Corresponding

potential

used in

band gap or Bragg reflection with a reciprocal vector g, the perturbing

a

energy. Cohen,04)

Blandin

to

For

should

the band-structure

first

perturbation

of the per-

principles.‘12*13)

a perturbation

good one for calculating The

band-

which also allows us to make various estimates

of the band such

their

Al, Mg, Zn, Pb) the band

and the perturbations

has been explained

V(R, -

stability

on phase transformation.

from that of a free electron

turbing

to any lattice

which scatters an

electron from the state exp (ik . F) has a scattering

We then obtainol)

~~,jV(Ri - R,),

(4)

to

X(g)v(g).

exp [--iq

(Ri -

l

Rj)]E(n) d3q + C.C. (5)

[To obtain

(4) we have included

(2) and omitted

the term

presents

the energy

between

atoms.

equivalent

the term g = 0 in

i =.j

both structure independent

in (4), which

quantities.]

Equations

then

the Rudermann-Kittel

(4)

forces

(2) and (4) are exactly the energy of a metal.

We note that if we make the perturbing spin dependent,

are

Equation

as a sum over pairwise

ways of analysing

potential

equation

(2) and (4) include

spin-spin

interaction,tg) which

occurs in the magnetic rare-earths. For structures planes

we

Saada,o)

composed

entirely

of close-packed

may,

following

Blandin,

Friedel

analyse

equation

(2)

contributions

from pairs of close-packed a close-packed

into

planes.

and

Suppose we have

plane, the coordinates

of whose atoms

where pz is the component

are Ri = (p,, pL + lil), in the x direction perpendicular

to the plane

are vectors perpendicular

P1 + li, i.e. lying in the plane. of the plane.

and

to this direction,

The liL are the lattice vectors

We now have

I:i exp (--ip

exp [i(k + g) . r]

amplitude

Rj) =

how the total

i.e.

years it has been realised

(e.g. the alkalis,

tial&)

the

is to find out

valence

energy,

atoms Ri and Rj.

U, = U, +

ANALYSIS

problem

energy

of

1967

where

GENERAL

basic

15,

phases in a transition

from h.c.p. to f.c.c.

The

VOL.

. Ri) = exp (-iig,p,

-

iq,

. pI)

Here X(g) is the

structure factor of the lattice exp (--ig

S(g) = N-l&

. R,),

(1)

and w(g) is a form factor depending only on the atomic nature of the metal which is calculated transforming

the pseudopotential.

in v(g) the structure

dependent

by Fourier-

To second

order

energy of the metal

may be written CT, = where U,

+ &I

lfl(g)12 ~(d2~2Mfk7L

is the Ewald energyo5)

of planes

theory.

We may write

where E(g) is the energy-wavenumber Harrison.05)

Typical

cos[g, *(Pi, -

gl

t

characteristic forms

perform

of v(g)

two-

(7)

to PLn’)]

I’

‘m

E(q)

-cc

expCiq,. (P,, - p,,,)l dq,, (8) q = (e2 + g_L‘-Y2. If we stack planes in ABC positions plnf can take the values +d two

only enters the formula through IS( or ]S(q)]2 and we may expand this quantity as a double sum over

from

and

illustrated

of the lattice

the

we get(lng)

X,,,@,,?

(3)

continuous

The structure

and

over q,

where Onn, is proportional

faulted lattice) equation (2) remains the same except that the sum over g now becomes an integral over a g.

n’,

7

and E(g) are shown in the article by Heine and Weaire.os) In the case of an imperfect lattice (e.g. a

variable

and

17, = u,

2

reciprocal

(2) in terms of contributions n

integration

and s2(g) and f(g)

E(g) = Q92ez(s)f(s) by

]X(q)]2 in equation pairs

of atoms per unit area in

are the two-dimensional

of the plane (see Fig. 1). If we now analyse

(2)

arising from the energy denominators

in the perturbation

considered

vectors

dimensional

u,

are functions

where co-r is the density the plane and g,

in Fig.

planes

positions

(A-A,

@&?I

p,,,).

-

1.

When

are in what B-B,

pin -

we

C-C)

When pin -

then pLLn-

where d is the vector shall

and

plnf = 0 the call equivalent

(8) has the value

pint = +d the planes

HODGES:

STABILITY

AND

AXIAL

RATIO

1789

TABLE 1. Fraction c,, of mth nearest neighbour planes in eqmvalent positions

h.r.p. Sm d.h.c.p. f.c.c.

. A atoms x 0 atoms 0 C atoms

.. EF (a)

Jq:

structures

in concerned,

positions

(A-B, p,,,).

between

the important

is the energy required from

positions

keeping

equivalent

to

the interplanar

does

0

As far

energy or ion-ion

been evaluated(lg) decreasing

a

p,,,)l Q,

-

contribute

interaction

to

4.

(9) As

of the

between

planes.

that in (9) we are only

concerned

having smallest non-zero

magnitude.

For

if the quantity planes

will

these to

be

positions.

shall assume that when the different

cos

(gl

. d) =

inside the integral

tend

inequivalent,

g,,

phases

and

-4

is negative

in equivalent, In the following comparing

the

if positive analysis we

the energies

we do so at constant

of

density

and axial ratio ; we also assume that the interplanar spacing within one phase is constant Appendix

(See

Let

c,

planes

be the

for

a discussion

fraction

(separation

mh)

and equal to h.

of

these

of mt” nearest in

equivalent

points.)

neighbour positions.

Then U, may be written

u, = m=l f 2

Tnz, [Q&h)

i

1 1 0

0 0

t 1

” 0

c9 0

:,

0

1

-

2

u,a =

crnb)$bh).

(C,a -

(11)

c,n$(mh)l.

1 we have tabulated

the values

contribution

to

structure

term.

contribution

Nevertheless,

has decreased

of c,

with.

Note

This is a result which

4(h)

dominates

is large the band

for m > 2 the Ewald by such a large factor

(350 in the case of Mg)os) as to be negligible compared to the band

structure

term.

An

explicit

form

+(z) may b,P g’oven b y using an approximation becomes valid at large values of x.(l) d(r) Cc 5+ sin [rti(4kp2

of

which

It is

-

gL2)]

(13)

4(z) K x-2

exp

[x1/(g_L2

-

4kp2)]

(13)

if IgLl > %P For all polyvalent

metals (e/a > 1.14) the first form is

the appropriate long-range We

oscillating

have

planar

one, and we see that it is a fairly

given

interaction

theory.

force.

here a derivation using

Nevertheless

multiple

scattering

that

the

analysis

true

to higher

approach into

orders

(9).

perturbation

it can be shown, such

of

interactions

remains

though,

of

Thus the fact that the rare-earth

cannot

electron

metalsoO) owing to the presence

be

considered

does not necessarily

DEVIATIONS

the

is no longer given

metals

into interactions

using

Bennemann,(20)

of perturbations

course, the form of the interaction by equation

of the inter-

second-order

to

be

nearly-freeof d-bands

mean that the general analysis

between planes is invalid. OF

AXIAL

RATIO

FROM

IDEALITY

In this section we shall use the method of interactions between planes to study the c/a ratios observed

[c,@,(mh)+ (1 - c,P,i(mh)l =

0

a

It may also be shown’l*ls) g,

:

2k, > lg;l

to q5; this has

of the distance

with the six

0

and

and found to be an exponentially

function

i

if E(q)

s --m

not

0 0 0 1

Ewald

constant.

result it is possible to include the contribution Ewald

we

+m

exp [%,(p,, =

1 3 & 0

and the negative and completely

to move

spacing

~=cDi-~~cc~,l[cos(gl.d)-l]

g,

of the

inequivalent

This is

Clearly,

C8

0 0 0 0

that cr is zero for all these phases.

A-C,

quantity

have to consider planes

c;

for the four structures we are concerned

close-packed

pair

of

cg

In Table

etc.) and (8) takes the value Qi(pZn in energy

cg

m=l

(b)

difference

cq

C2b -

-I-

.

FIG. 1 (a). The three types of close-packed plane A,B,C shown in projections and the vector d (b). The two-dimensional reciprocal lattice of a close-packed plane, showing the origin 0, and the relative orientation and scale to the planes in (a).

as the

c3

The energy difference between two structures a and b

.

are said to be in inequivalent

cz

ma- be writ,ten

2n

.

.

c1

in

(10)

various

argument,

closepacked which

is

structures. essential

to

The the

following subsequent

analysis, was first suggested by D. Weaire.@l)

ACTh

1790

‘i

!

METALLURGICA.,

VOL.

15,

1967

in such phases to be of two sorts: either a small deviation. where the corresponding cubic phase is ideal (f.c.c.)! or a large one where the cubic phase is distorted. We believe that Mg and the rare-earths are of t’he former kind, whereas Cd and Zn are of the labter. Two rare-earths, La, Ce, do indeed show true f.c.c. structures. This is the argument first given by D. Wcaire.(21) Let us suppose we are dealing wit&hthe fir& kind of system and write the axial ratio c/a = T and the ideal ratio. 1.633 = ro. Equation (l-1) may be expanded

&al (0)

(b)

Fru. 2. The two types of U vs. c/a CW’YRSfor the rhomboIn (a) the f.c.c. structure is stable; hedral structure. in (b) a distorted structure with non-ideal c/a ratio.

c’ will have minimum energy when r-r,=

When we vary the c/a ratio to determine the value at which the energy is a minimum we must do so at constant density for the reasons given in the Appendix. If we take a true f.c.c. structure and distort it by introducing extensions or compressions d,, d,, d, along the principal axes x, y, x then by cubic symmetry we have to first order, 6U is proportion to (dz + d, + cl,). But if the volume is constant d, + d, + d, = 0. Therefore a cubic structure has a stationary value of the energy with rezzpect to volume conserving distortions, whether or not it is stable with respect! to them. As we introduce a rhombohedral distortion to the f.c.c. structure the form of the energy vs. c/a curves must be one of two types (see Fig. 2). In the first type the f.c.c. structure with ideal c/a ratio will be stable, and in the second type a rhombohedral structure with an appreciable distortion, such as the cc form of Hg will occur. For any phase other than f.c.c. we may write using (IO),

u= - f

bd(mh) + Uf.c,r..

(14)

We note that $(h) does not contribute to the energy difference, but that O{(h) does contribute to the T structural energy of 0 f.t.c,, and that #(2&j, #(3h) are in general smaller than (9Jh). As there is no symmetry argument equivalent to that for the f.c.c. for any of the other close-packed phases we are considering, the sum over $(mh) in (14) will give a small term, linear to a first approximation in c/a, driving the axial ratio of the particular phase away from that of the corresponding cubic or distorted cubic phase. Thus we should expect deviations of c/cz from ideality

[>I;,’

in:.,[v] AC,,,

1 To

( 16) (Note that ~#~~r is not simply related to ~~/~x ; also that there is no reason why, if a particular (5(z) = 0, &#J/&should also be zero). If WC were now to assume that the t’erm nz.= 2 in equation (16) dominates those terms with higher m values t,hen we should have r-

ru K Ac,

(17)

where AC,? from Table 1, is .&, $, 1 for the d.h.c.p., Sm and h.c.p. phases respectively. At a transition between phases we should expect to observe a discontinuity in c/a, such that deviations r - r0 on either side are in the ratio of the corresponding AC, values. In Fig. 3 we have plotted r - r0 against AC, for the phases d.h.c.p., Sm, h.c.p. appearing in the rare-earth alloy systems studies by Harris.(‘) Since all these phases appear within a narrow composition range, we have plotted their axial ratios all on the same graph. In Fig. 4 we show two of the plots of Harris from which these results are taken and also indicate the c/a ratios taken to represent the different phases where c/a varies appreciably with concentration. In the system Y-Ce there is 8ome uncertainty in the value of C/Gto take for the h.c.p. phase. None of the graphs are ent,irely flat in the region of the transition; this may be due in part to a smoothing out of the discontinuity (e.g. the presence of a randomly faulted structure). On the other hand, it is surely not correct to take the c/a ratio of pure Y in the Y-Ce system, as its very low c/a ratio is associated with its very stable

HODGES:

STABILITY

AND

AXIAL

RATIO

1791

.06 T 1 615

c

Atomic

“1. Ce (a)

1.605M 1.600-

FIG. 3. Plots of deviations of the axial ratio from ideality for the four alloy system Gd-Ce, Gd-Pr, Y-Ce, Y-Pr. For the last two systems the c/a ratio of the h.c.p. phase is uncertain. The limits of uncertainty are shown in Fig. 4 for the Y-Ce system. 0

h.c.p.

structure

addition

which

50 atomic

only

transforms

y0 Ce.

after

the

We have indicated

the

limits between which one might take these c/a ratios. We

see that

for the two

systems

where there is not much

Gd-Ce,

uncertainty

to take, there is remarkable

Gd-Pr,

which values

agreement with equation

(17). As regards the Mg and silver alloys, the c/a plots in Ref.

(3) may

continuities

be interpreted

corresponding

It is noted that although

obeyed

the change in c/a is of the

transforms

cases is the proportionality

to the

and co-workers reported.

sign (i.e. to decrease

ideal when h.c.p

out dis-

to the transitions

d.h.c.p. structure Drickamer expected

as smoothed

the deviation

to d.h.c.p.),

from

only in two

rule ( 17) evenapproximately

These are the Mg alloys with e/a > 2 and

the Ag-In only system

II alloy

(e/u = 1.57).

This latter is the

with a well defined transition (r (r -

The proportionality

r,)d.h.c.p.

and here

= o 48,

To)h.c.p.

’

observed

by Drickamer

and

in those of Ref. (4) and (5) the c/a ratio of the d.h.c.p. phase is often surprisingly of appreciable value of r -

60

40

Atomx

%

60

100

Ce

(b)

FIG. 4. Two of the plots given by Harris (a) Cd-Ce (b) Y-Ce. The phases are indicated:- 0 h.c.p.; n Sm; x d.h.c.p.

Why this should be so is not at present clear, though, of course,

several

approximations

in deriving equation In

view

of these

approximations

agreement with equation Gd-Pr some

is somewhat special

important electron

reason

remarkable.

There

m > 2 are

metals because level.

free-electron

&mh),

interaction

of

may

for the rare-earths.

be not The

by no means

nearly-free-

of the presence

of d bands

However,

rule does not necessarily interplanar

the degree

why

are of course

at the Fermi

been made

(17) for the systems Gd-Ce,

in this problem

rare-earths

have

(17).

the proportionality

depend on the form of the derived

perturbation

from

theory section

given the

the

nearly-

in (8) ; as

mentioned

in the previous

interaction

can include higher orders of perturbation

interplanar

theory.

rule would indicate 0.5.

In the other systems

20

close to ideal, in spite r0 for the h.c.p. phase.

STABILITY We

OF

CLOSE

PACKED

have seen in the previous

STRUCTURES

sections

how the

method of interactions between planes can explain in certain cases the deviations of c/a from ideality

ACTA

1792

METALLURGICA,

in close-packed phases. We shall now apply the same method to studying the conditions under which one would expect the appearance of complex intermediate phases like d.h.c.p. and Sm. As can be seen from equation (12), &2h) is normally appreciably greater than #(3h), $(4h), etc. Under these conditions we see from equation (10) that either the h.c.p. phase will be stable [if $(2h) is positive] or the f.c.c. phase will be stable [if #(2h) is negative]. This is because these are the phases with the largest and smallest possible values of c2, and getting the largest contribution from $(2h) to the binding energy will outweigh all considerations of getting energy from &3h), #(4h) etc. This is then the reason for the common occurrence of h.c.p. and f.c.c. phases. Suppose now that we have an h.c.p. phase with &ah) large and positive, and that alloying or compression causes &2h) to decrease through zero. From the above argument we see that this will ultimately result in a transition to f.c.c. When r$(2h) is small, however, we have the possibility of observing inte~ediat,e phases since c.&2h) is no longer the dominating term in the energy. Let us for the mon~ent ignore the Sm phase. From Table 1 and equation (11) we see that u d.h.r.p. u f.r.c. -

-

%c.,.

%h.c.p.

= =

t+W)

#2h)

+

:yW)

- +(3h)

+

. . .

(18)

+ +(4h) + . . . (19)

Since from equation (Ii?), I will be small, we may consider the energies of the d.h.c.p. and h.c.p. phases to become equal when #2h) = 0, and beyond this point d.h.c.p. will be the more stable of the two phases. However, if when &2h) = 0, -$(3h) + &4h) + ... is negative then the transition to d.h.o.p. will not occur since f.c.c. will have already become the most If however +(3h) -+ #(4h) + . . . stable phase. is positive then f.c.c. will only later become the most stable phase when $(2h) becomes significantly negative to make the right hand side of equation (19) negative. Thus the condition for the d.h.c.p. phase to appear between the f.c.c, and h.c.p. phases is that -#(3h)

-j- &4h)

+

...> 0

(20)

when &2h) = 0. A similar analysis may be carried out for the Sm structure. The general conditions governing the appearance of these structures may most easily be seen by examining Fig. 5 (see below). However, we may say that if condition (20) is satisfied then the

VOL.

15,

1967

hcp ._ . . . . . . . .

Sm

_.-__._.

dhcp

--___

tee

FIG. 5. Piot of theenergy U-Uh.e.a. against ~#(Zh)for the phases f.c.c., d.h.c.p., Sm and h.c.p., under condition where both intermediate phases appear. The slopes of the lines are in the ratio 1: i : j: 0 corresponding to the coefficients of 4(2h), and the interceptson theaxis #(2h) -= 0 are given by the 1.h.s. of (20) and (21) for f.c.c. and Sm respectivety.

condition for the Sm structure to appear is that t’ SIB -

zid.h.d,.

i.e. that

<

0 when +(2h)

$$(4h)

-

+$(5h)

= 0 +

. . . < 0.

(21) We also see from Fig. 5. that if, under these conditions the 1.h.s. of (20) is less than 3 times the 1.h.s. of (21) in magnitude the d.h.e.p. phase will no longer appear. In this analysis we have assumed that the expressions in (20) and (21) do not change sign over the small range of &2h) where these complex structures can appear. In fact the sign of the asymptotic form of +(x) given in equation (12) depends only on the electron to atom ratio(l) and should therefore not change in the intra rare-earth alloy systems and in systems subjected to pressure. Since, however, we observe phase transitions in these systems it is clear that (12) is only an approximation. It is however an approximation which becomes more accurate as x increases and we should expect that the expressions (20) and (21) will not change sign if they are not too small. In Fig. 5. we have presented plots of the energy of the various phases against ~(2~), assuming the expressions (20) and (21) remain roughly constant. From these plots one can visualise the conditions given above for the appearance of the phases, and work out approximately what their relative ranges of stability are, assuming $(2h) varies linearly with composition or pressure. It would be interesting to see if the asymptotic form (12) gives any agreement with monitions (20) and

HODGES:

STABILITY

AND

AXIAL

1793

RATIO

TABLE 2. Tabulation of m-% sin [mh y’(4k~J - gL2)] = x$(A) using equation (12), and the contributiona to expressions (20) and (21) e/a = 1.26 m 3 4 5

e/a = 1.5

e/a = 2.0

-

e/a = 3.0

a+

(20)

ab

(20)

=+

(20)

c+

(20)

(21)

-0.11 0.00 + 0.04,

io.ll 0.00 0.00

+0.11 -0.05 + 0.02

-0.11 -0.05 0.00

-0,118 +O.O6 + 0.02

im0.08 -t 0.06 0.00

- 0.082 -0.004 +0.025

+0.082 - 0.004 0.000

- 0.003 - 0.008

--

-

(21) for those electron-atom

ratios at which d.h.c.p.

and Sm structures

have

been

observed.

therefore

the values

of &3h),

from

tabulated

(12) for these e/a. ratios

should use a form of equation down exponentially

For

the

definite

d.h.c.p.

There is

impo~ant,

metal lattice one

position

of

and

2.).

prominent

of

its

zero

reciprocal

The values

structures.

From

down by an unknown

Weaire(l*J

a trend

beyond $( 5h) on the results.

we see that

condition

(20)

there for

is

those

elements

the

(9), and that

shape

qualitatively

with

(12) is not always

to equation

of q becomes

(12) which is damped

structure

agreement,

by equation

function

factor ; we have therefore not continued and can only comment

form given

approximation detailed

at very large distances.

in Table 2, should be damped

have

$1(4h), etc.

(Table

reason to believe that in a perfect

We

pseudopotential

vectors

of the same valence,

potential vector

of view

away of

one

from

and f.c.c.

of Heine

and

of a series of

e.g. the rare-earths,

a prominent

reciprocal

towards

of

lattice another

structure.

Au-Gac5)) ; 2, (Mg and its al10ys(~)) ; 3, (rare-earths and

that in the trivalent

alloys).

from f.c.c. and the distorted f.c.c. structures of Al, Ga,

(Ag-In

There is however alloys

with

definite

electron

and Ag-Sbc3)).

to

disagreement

atom

ratio

for

~1.5,

The values of 4(x) in Table 2

might

one

phases with electron to atom ratio of 1.26, (Au-in(4),

those

one

to

of the zero of the pseudo-

structure,

Thus

a

the

relative

of the h.c.p.

in the structure

is caused by a movement

as

v(q)

in particular

maximum

the point

a good here the

In to the h.c.p. structure the pseudopotential

of Tl is due to the zero of

do not, predict a clearly defined stability or instability

from t*he main concentration weight

(see Ref.

tentatively

we cross the rare-earths from left to right is due to a

suggest that the trend towards

to a definite

probable

that,

interactions

between planes at quite large separations

similar

in condition

(21) the deviations

pseudopotential.

[the

surface underlies Indeed, occur

from

as we are dealing

of the Fermi surfaces

a sphere will become

assumption

of

a spherical

the derivation

of equation

very

rare-earths

have

been

movement

transition

( 12)(1)].

earths.(@)

explained

by

Williams, Loucks and Mackintoshc22) as resulting from

outwards We note

system(24) the application

Fermi

some of the long range spin structures which in the

f.c.c.

to h.c.p.

This would be explained

point of view by the movement lattice

vectors

thus reverses the movement

structure.

v(q)

c direction,

distance

in the

should not derive its stability in the same

way as the spin waves do.

Such an explanation

of

to that phases

in the rare-earths have

been

to using an interplanar

rare-earths

with a long range

shape of the Fermi surface and approximately

by the equal

to the repeat period 9h of the Sm structure.

Calcula-

tions by Keeton and Loucks(23) indicate between 9h and 10h for this interaction.

a period

VVe shall

end

this

se&ion

by

a

few

remarks

unaltered. lattice

depend

have

these

quantitatively. explanation

observed yet

ideas

the

while

Pressure

of the zero of vectors.

The

of

being a sphere. A calculation

the

trends

no intermediate

in the former. been

cannot

It should

critically

in that

calculated yet

be noted in

be

As no for the

confirmed

that the above

structure

does

not

on the shape of the Fermi surface has been

performed

for Mg(25) to

see if mo~~en~ent of the zero of v(q) with respect to the

concerning the origin of the change in sign of qb(2hf required to produce the sequence in structures

reciprocal

vectors on compression

transition

observed

from h.c.p. to f.c.c. energies@) indicates

was used, with values Heine and Abarenkov

Previous work on stacking fault that for $(2h) the asymptotic

reverses

In-T1 f.c.c. t,o h.c.p. transition’24) is however dissimilar

pseudopotentials

interaction

the alloy

from the above

outwards

to the reciprocal

the occurrence of the Sm structure would be equivalent part, which oscillated with a period determined

of

the material,

the zero of v(q) remains almost relative

zero

outwards of reciprocal

as we compress

of parallel regions of the Fermi surface with its large repeat

the

that in the In-T1 of pressure

of the h.c.p.

There is no reason why the

of

h.c.p. as

just as it does in the rare-

the existence Sm structure,

structural

One might even more

with

to come

It is very

of the rare-earths

of the h.c.p.

18, Fig. 4).

here.

calculation

important

conclusion

suggest

of Tl having moved outwards awa y

for the Sm structure.

It would require more detailed

tentatively

series Al, Ga, In, Tl the trend

would give us the

by Drickamer.c3j of v(q) model

Equation

(2)

calculated from potential.(12*13)

the No

ACTA

1794

transition

to d.h.c.p.

anything

like

Drickamer . give

good

the

or to f.c.c.

compression

Although

the

agreement

was positive

was observed

value

values

with

METALLURGICA,

observed

at by

of

&3h),

$(4h)

condition

(20),

cj(2h)

and

advice and to Mr. D. Weaire for much of the argument of the c/a ratios.

to thank Dr. A. Blandin useful discussions, to reproduce

I also wish

his diagrams.

Research

Council

comparing

grant,

which is gratefully

acknowledged.

approximations

introduced

of phases

at the same

spacing within one structure to be constant. consider the volume

c = U,(V) where U,(V) We

have

+ C,(V)

is a large, structure

up

structure

to

now

dependent

metal is almost

only

(Al)

independent

considered

U,(V).

entirely

at V,.

of

v Substituting

we

U may

V,as

V’o)U,‘.

(A.2)

given by

8, = - U,‘/U,“.

(A3)

(A3) into (A2) we get

U( V) = U,( V,) + In general

which

Vo)2U1”

+ (V -

V is

V of the

U,(v),

U = CT,(V,) + U,( V,) + $( v -

volume

small,

The energy

in the neighbourhood

The equilibrium

term.

the

The volume

fixed by

has a minimum

be expanded

REFERENCES 1. A. BLANDIN, J. FRIEDEL md G. SAADA, Proceedings of the Toulouse Conference on Dislocations, J. Phus. C3. 128 ” (1967). 2 C. H. HODQES, Phil. Mag. 15,371 (1967). 3: E. A. PEREZ-ALBUERNE, R. L. CLENDENEN, R. W. LYNCH and H. G. DRICKAMER, Phys. Rev. A142, 392 (1966). 4. S. E. R. HIXOCKS and W. HUME-ROTHERY, Proc. R. Sot. 282, 318 (1964). 5. C. J. COOKE and W. HUME-ROTHERY, J. less-commons Metals 10, 42 (1966). at High Pressures, p. 478, 6. A. JAYARAMAN, Physics ofSolid Academic Press. New York (1965). 7. I. R. HARRIS, 6. C. KOCH and 6. V. RAYNOR, J. lesscommon Metals 11, 436 (1966). 8. A. JAYARAMAN, Phys. Rev. A139, 690 (1965). 9. Y.-A. ROCHER, Adv. Phys. 11, 233 (1962). 10. V. HEINE, in: The Structure and Properties of Metals, Cambridge University Press (to be published). in the Theory of Metals, 11. W. A. HARRISON, Pseudopotentials Benjamin, New York (1966). 12. V. HEINE and I. ABARENKOT, Phil. Mug. 9, 451 (1964); 12, 529 (1965). 13. V. HEINE and A. 0. E. ANIMALU, Phil. Mag. 12, 1249 (1965). and Effects in Concen14. M. H. COHEN, Alloying Behaviour trated Solid Solutions, p. 1, Gordon & Breach, New York (1962). 15. W. A. HARRISON, Phys. Rev. 129,2503 and 2512 (1963). 16. L. J. SHAM, Thesis (unpublished). University of Cambridge (1963). p. 50, Benjamin, 17. A. BLANDIN, Metallic Solid SOlUtiOn8, New York (1963). 18. V. HEINE and D. WEAIRE, Phys. Rev. 152, 603 (1966). 19. C. H. HODOES, Phil. Mag. 15, (1967). au: K. M. BENNEMAN, Phys. Rev. Al& 1045 (1964). 21. D. WEAIRE, private communication (1966). 22. R. W. WILLIAMS, T. L. LOUCKS and A. R. MACRINTOSH, Phys. Rev. Lett. 16, 168 (1966) 23. S. C. KEETON and T. L. LOUCKS, to be published. 24. R. W. MEYERHOFF and J. F. SMITH, Acta Met. 11, 529

Let us

V first of all. The total energy of

the metal may be written(1°s15)

suppose

Note added in proof. Since submission of this article, work by Gschneider and Valletta (to be published) has come to the notice of the author concerning a correlation between the ratio of the 4f shell radius to the metallic radius and the crystal structure in the rare-earths. In simple metals with s - p type conduction bands the zero of the pseudo-potential is closely related to the core to metal radius ratio.(27) Gschneider’s explanation of the trends of structure with pressure and atomic number is thus similar in principle to that suggested above. It should be mentioned that if we are considering the d and f shells it is more meaningful to talk in terms of phase shifts and t-matrices than in terms of the pseudo-potential.

(1963)

the

the energies

and Professor J. Friedel for

and Dr. I. R. Harris for permission

This research was carried out during the tenure of a Science

we discuss

volume and c/a ratio and by assuming the interplanar

to Dr. V. Heine for guidance

the analysis

1967

APPENDIX

Here by

ACKNOWLEDGMENTS

I am grateful

15,

25. C. H. HODGES, unpublished. 26. W. B. PEARSON, Handbook of Lattice Spacings and Structures of MetaZs,p. 129, Pergamon Press, Oxford (1958). 27. N. W. ASHCROFT, Phys. Lett. 28,48 (1966).

and did not appear to reverse sign until

very much higher values of the compression.

behind

VOL.

we may

U,( Vo) -

Wz’12/Uz”.

(A4)

assume the third term in (A4)

to be smaller than

U,,since it depends on the quantity U,. Thus the binding

square

of the small

energy

gained by allowing the structure to relax from its correct

equilibrium

volume

small structure dependent The same argument

is smaller

energy itself.*

holds as regards the c/a ratio,

at any rate for the substances Here the role of

keep

the

small

It is anyway only

obtain

clear from

spacing

and that &2h),

perturbations

deviations

Again,

plays the role of

interplanar

within the one structure, produce

played by the second

(14) respectively.

we may assume that @Jh) to

we have considered.

U, and U, are

and first terms of equation tending

V,to

than the

$(3h)

on the equal

symmetry from

equal

U,,in

constant only

spacing.

that we should spacing

for the

Sm structure. The arguments fact that changes

given

here are borne

in volume,

out by the

c/a ratio on changing

structure are indeed small. No deviation from equal spacing appears to be reported in the literature.(26) * For instance, in the rare-earth metals volume changes of typically +0/O occur on transforming phase. Using the experimental value of the compressibility as a rough value of U,“, we have estimated that volume changes of this order would contribute 1O-6 N 1O-6 Rydberglatom to the difference in energy between the phases. This is much less than typical values of AU, encountered in metals (1O-2 N 1O-3 Rydberg] atom).