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V ol . 1 0 1 ( 2002)
A CT A P HY SIC A P O LO N IC A A
No . 5
P r oceedi ng s o f t h e Sy m p o siu m o n Syn chr otr on Cr y st all ogr ap hy, Kr y n ica, Po l an d 200 1
P h ase T ra n sfor m at ion s in B in ar y A llo y s
u n d er H igh Pre ssu re
V .F. D egt y are v a
I nst i t ut e of Sol id State P hysi cs, R ussian Aca dem y of Sciences
Cherno gol ov ka, Mo scow di stri ct, 142432, R ussia
H igh-pressure stu dies of bi nar y alloy systems displ ay some speci Ùc features connected w ith comp osition referring as a variable parameter . In a
tw o-comp onent system under pressure there are observed transf ormations ,
w hich do not occur in a one- comp onent system. In particular, there is a formation of a single phase from a mi xture of tw o phases or a decomp ositio n
of a unif orm phase into two phases of di˜erent comp ositi ons. A lloys of I n
and Sn w ith neighb ouring elements give us examples of these kinds of transformations. Studies on binary alloys w ith applicati on of pressure pro vide
exp erimental evidences that the alloy comp ositi on or electron concentration is one of the main factor for structural phase stability follow ing from
H ume { Rothery rule.
PACS numb ers: 61.50.K s, 62.50. + p, 61. 66.Dk
1. I n t r o d u ct io n
Hi gh- pressure studi es a re mai nly focused on elements or on com p ounds al rea dy exi sti ng at a mbi ent pressure, e.g. on one-comp onent or qua si-one-comp onent
system s. Investi g ati ons on bi na ry and mul ti -comp onent system s under pressure are
l arg ely con Ùned to gi ven al l oy com p ositi on. For a two -comp onent system along
wi th two therm odyna m i c pa ra meters, pressure and tem p erature, one ha s to consider an addi ti o nal vari abl e pa ra meter | com positi on. Thi s o˜ers f or a bi nary system an addi ti ona l degree of f reedom i n pha se equi l i bri a and al l ows the o ccurrence
of pha se tra nsf orm ati ons under pressure, whi ch do no t occur i n a one-com p onent
system .
A bi na ry com p ound of a stoi chi om etri c comp ositi on i n hi g h-pressure studi es
i s usual l y appro ached to a quasi- one-com p onent system, however thi s consi derati on
coul d b e appl i ed wi th certa i n restri cti ons. It i s often observed tha t when accordi ng
to therm odyna mi c equi l i bri um condi ti ons a bi nary com p ound shoul d decom p ose,
( 675)
676
V .F. Degt yare va
the tra nsf orm ati on does no t occur b ecause of ki neti c hi ndra nce of di ˜usi on pro cess.
The observed state of the sam pl e i s theref ore metasta bl e. W i th an i ncrease i n
pressure the melti ng tem p erature usual l y i ncreases to gether wi th the di ˜usi on
ba rri er tha t hi nders the atta i nm ent of the equi l i bri um sta te. Thi s b ehavi our shoul d
b e ta ken i nto account when one considers pha se tra nsfo rm ati ons of sto i chiom etri c
comp ounds under pressure.
At Ùrst gl ance one shoul d expect tha t appl i cati on of pressure to bi nary al l oy
system s wo ul d pro duce a m ore com pl icated pi cture i n pha se equi l i bri a compa ri ng
to tha t at the am bi ent pressure b ecause of :
| p ol ym orphi c tra nsiti ons of consti tuents ;
| pha setra nsiti ons of ambient- pressure interm ediate pha sesand comp ounds;
| decomp ositi on of ambi ent- pressure com pounds;
| f orm ati on of new i nterm edi ate hi gh-pressure pha ses a nd thei r tra nsf orm ati on on further pressure i ncrease.
System ati c studi es on bi nary al l oys under pressure were ini ti ated by Po nya to vski i wi th the \ pressure quenchi ng " metho d (R ef. [1]) and conti nued i n sit u
under pressure (R efs. [2{ 8]). These studi es show tha t the com mo n pi cture of T { x
pha se equi l ibri a fo r bi nary al l oy systems can b e sim pl iÙed by i ntro duci ng an a ddi ti ona l vari abl e | pressure. Three- di m ensiona l T { P { x pha se di agra m s f or hom ol ogous bi na ry system s turned out to be sim i la r to each other, but shifted al ong the
pressure axi s. Mi ssing at am bi ent pressure i nterm edi ate pha ses can app ear i n a
bi nary al l oy system under hi gh pressure.
Under pressure the gro up IV and V elements o f semi conducti ng/ semi meta l l ic
typ e tra nsform to a metal l i c sta te. In the bi na ry al l oy system s of these el ements
wi th the neighb ouri ng m etal s the pha se equi li bri a evol ve under pressure f rom
semi conduct or /semi met al{met al type to met al {met al typ e. The m ai n speciÙc features of these chang es are as fol l owing :
| extensi on of sol i d sol uti ons based on consti tuents or ambi ent- pressure
i nterm edi ate pha ses of a metal l i c typ e;
| decom p ositi on of stoi chi ometri c com p ounds or i nterm edi ate na rro w-ra ng e
pha ses;
| form ati on of new i nterm edi ate hi gh- pressure pha ses wi th m etal l i c typ e of
b ondi ng ;
| occurrence of pha se sequences along val ence electro n concentra ti on di spl ayi ng a decrease i n packi ng density and coo rdi nati on numb er;
| determ i na ti on of the al l oy pha se sta bi l i ty predo mi na ntl y by a f acto r of the
val ence electro n concentra ti on i n spiri t of the known Hum e{ Rothery or el ectro n
pha ses.
Studi es on bi nary al l oys of s p m etal s wi th the rel a ti vel y sim pl e electro n conÙgura ti on are of special interest. A bi na ry al l oy of the elements nei ghb ouri ng i n
the p eri odi c ta bl e wi th m i ni m al di ˜erences i n ato m ic sizes and electro negati vi ty
can b e considered as a \ m odel l i ng element" wi th the va lency a s a vari able para m -
P hase T ransfor mat i ons i n Bi nar y A l l oys
.. .
677
eter. Thi s system i s very suita bl e for theo reti cal ana l ysi s on the structure sta bi l i ty .
The hi gh- pressure pha ses obta i ned i n In and Sn based a ll oys gi ve exp eri menta l
evi dences for Bri l l oui n- zoneÀ Fermi -sphere (B Z{ FS) Ùtti ng as a cruci al fa cto r of
the pha se sta bi l i ty for these nearl y-f ree-el ectro n m etal s. W i th thi s respect hi gh
pressure i s used a s a to ol to anal yse pha se sta bi l i ty i n m odel l i ng meta l s wi th a
\ vari abl e valency" . These studi es are i m p orta nt i n rel ati on to electro n tra nsfers
under pressure observed i n several gro ups of el ements such as al kal i and al kali- earth
el ements, l antha nides and acti ni des.
In thi s pa per som e features of pha se sta bi l i ty i n bi nary al l oys of sp m etal s
wi l l b e o utl i ned on the base of recent hi gh- pressure studi es p erform ed in di amond
anvi l cell s wi th the use of synchro tro n ra di ati on (R efs. [2{ 8]).
2. F o r m at i o n o f n e w i n ter med iat e p h ases i n b i n ar y a l l oys
u n d er p r essu r e
Pha se equi li bri a i n a system of C comp onents existi ng i n P pha ses are determ i ned by the G i bbs pha se rul e: at given pressure and tem p erature as vari ables,
the numb er of degrees of f reedom, F , i s F = C À P + 2 . A two -comp onent system
theref ore has one degree of freedom m ore tha n a one-com p onent system . W hen the
pressure i s vari ed at a Ùxed tem pera ture, i n the system wi th C = 2 three pha ses
can coexist in equi li bri um. Thi s al lows tra nsf orm ati ons fro m a two - pha se m i xture i nto a sing le-pha se sta te a nd i nversely | a decomp ositi on of a sto i chiom etri c
comp ound or an i nterm edia te pha se i nto two pha ses of di ˜erent com p ositi ons.
2. 1. T he Sn-based bi nar y al l oys wi t h In and Hg
A simpl e hexagonal (sh) pha se exi sts at am bi ent condi ti ons onl y i n Sn-ba sed
al l oys wi th In and Hg i n qui te na rro w com p ositi on regi ons corresp ondi ng to the
el ectro n concentra ti on of ¤ 3 : 8 el./ ato m [9, 10]. Pha se di agra ms of In{ Sn and
In{ Hg at no rm al pressure are gi ven i n Fi g. 1 to gether wi th pl ots of the pha se
sequences observed under pressure [4{ 7].
The i nterest to hi gh- pressure studi es of these al l oys Ùrst ari ses fro m the
o ccurrenc e of sh pha se i n pure gro up IV elements Si and G e under pressure, whi l st
i t i s not fo und i n pure Sn. The l ig ht gro up IV elements Si and G e show some
comm on features in thei r hi gh- pressure tra nsform ati ons wi th the general trend:
di amond ( cF 8 ) ! whi te ti n ( t I 4 ) !
sim pl e hexagona l ( hP 1 ) !
close pa cked
structures ( hP 2 ; cF 4 ) [11{ 15].
Besides these hi gh- sym metry structures, al so som e i nterm edi ate l ow-sym m etry structures ha ve b een observed. The whi te t i n pha setra nsf orm s to sh thro ugh
an ortho rho m bi c I mm a structure, oI 4 [16, 17]. The tra nsform ati on f rom sh to hcp
o ccurs i n Si thro ug h a l ow-sym metry i nterm edi ate pha se, Si -VI [12], determ ined
as oC 1 6 , space gro up C mca [1 8], recentl y found al so i n G e [19].
678
Fig. 1.
V .F. Degtyareva
Phase sequences along the comp ositi on in the H g{Sn, I n{Sn and Ga{Sn alloy
systems. A mbient pressure phase boundari es are based on the data of [9, 10]. T he stars
and triangles on the phase diagrams indicate the comp osition s of the alloys studied in
our w ork. T he plots below the diagrams
show schematically
regions of phase stabili ty
under pressure and the phase sequences along the comp osition.
The heavi er gro up IV element Sn di spl ays a di ˜erent structura l sequence
under pressure [20 ] wi th di am ond ( cF 8 ) ! whi te ti n ( tI 4 ) ! b ody- centered tetra gona l ( tI 2 ) ! bo dy- centered cubi c ( cI 2 ) , omi tti ng hP 1 and the rel ated di sto rted
structures .
Our recent studi es on the sh pha se i n Sn{ In al l oys show tha t sta rti ng wi th
a sing l e-pha se al l oy at am bient pressure we obta i n a two - phase mi xture on the
pressure i ncrease [5, 7]. Thus the sh pha se i n the In 2 0 Sn8 0 al l oy tra nsform s to
hcp and b ct pha ses abo ve 10 G Pa and rel ati ve am ounts of these pha ses i n the
al l oy chang es wi th pressure, so tha t onl y the hcp pha se exists ab o ve 25 G Pa . The
studi es on other comp ositi ons of In{ Sn al l oys nei ghb ouri ng the sh pha se (Sn wi th
10 and 30 at. % In) al low for observ ati on o f sing le-pha se Ùelds wi th b ct and hcp,
as shown i n Fi g. 2.
Hg { Sn al loy s wi th the sh pha se show sim i l ar b ehavi our to In{ Sn al l oys,
tho ugh i n thi s system the hcp pha se occurred at much hi gher pressures aro und
55 GPa (R ef. [4, 7]). The hcp pha se i n In{ Sn and Hg { Sn al l oys is a new i nterm edi ate
Phase T ransf or mat i ons i n Bi nar y A l l oys
Fig. 2.
A
P
À
x
.. .
diagram of phase regions for the In{Sn alloy system. Di˜erent
679
symb ols
on the phase diagram denote exp erimental p oints for the alloys studied, corresp ondi ng to
di˜erent
phases observed. T w o (or three ) symb ols at one point indica te a phase- mixture,
w here the corresp ondi ng comp osition of the phases is show n by hori zontal arrow s. Phase
comp ositi ons are determined from atomic volume considera tion . F or exp erimental details
see Ref . [7].
hi gh- pressure pha se i n Sn-al l oys found exp eri m ental l y. The sh-to -hcp tra nsform ati on was recentl y repo rted for the al l oy In 2 5 Sn7 5 by the gro up of Meenakshi and
co-workers [22]. These studi es p oi nt out to the regi on of sta bi l i ty for the hcp pha se
near 3.75 el ./ ato m. The sh-to -hcp tra nsf orm ati on was al so observed i n Al 3 0 G e7 0
al l oy under pressure, when both sh and hcp pha ses are i nterm edi ate hi gh- pressure
pha ses [2].
The b ct pha se i n Sn al loys i s rel ated to the hi gh- pressure b ct pha se i n ti n
(Sn I I) [20] and shoul d b e regarded as a soli d sol uti on ba sed on Sn I I. The sta bi l i ty
reg ion of the b ct sol i d sol uti o n extends to ¤ 1 5 at. % In and to ¤ 1 0 at. % Hg , and
these regi ons tend to narro w wi th the pressure i ncrease. An i nteresti ng feature of
these bct pha ses is a correl ati on between a degree of tetra gona l di sto rti on ( c= a
ra ti o) and the al l oy comp ositi on. The axi al ra ti o decreases wi th a decrease i n
al l oy electro n concentra ti on. The possibl e expl anati on of thi s correl ati on shoul d
b e accounted for by Bri l l oui n-zone{ Fermi- sphere Ùtti ng s di scussed i n pap ers [3, 4].
680
V .F. Degtyareva
2.2. T he sh phase i n the Ga{Sn al l oys
The G a{ Sn system has a pha se di agra m of simpl e eutecti c typ e at zero
pressure [9, 10 ] (Fi g. 1). Ho wever, i n the hom ol ogous In{ Sn system an i nterm edi ate
sh pha se o ccurs on the Sn-ri ch side (a ro und 20 at. % In) al ready at zero pressure
[9, 10]. Form ati on of a sim i la r pha se was expected i n G a{ Sn al l oy system under
pressure . Fro m the previ ous studi es on bi nary al l oys a \ hom ol ogy rul e" was found
for T À P À x di agra m s, whi ch says tha t the e˜ect of pressure i s often sim il ar to a
repl acement of one of the consti tuents by a heavi er el ement fro m the same gro up
i n the p eri odi c ta bl e [1].
Fig. 3.
2˚ =
Energy
9 : 63
mixture:
£
(E
disp ersive di˜ractio n spectra for the Ga 2 0 Sn80
d = 73 83
b ct Sn I I w ith
b ct- Ga w ith
keV pm). T he pattern
a =
a = 265 : 7( 4)
3 55 :1 (2 )
pm,
c =
pm,
407 : 6(6)
33 0 :0(3)
pm,
c =a =
pm (p eaks
hk l
Ga). A fter pressure decrease
c =
to 11. 4 GPa and heating at 150 C for 1.5 h the pattern
£
w ith
a =
291 :5(2)
of bct- Ga with
a =
pm,
c =
270 : 7(2)
266 :4(12)
pm,
pm,
c =
c =a
410 : 2(6)
=
0 : 929
F or exp erimental
details see Ref . [7].
0 : 929
(p eaks
hkl)
and
at 23. 8 GPa corresp onds to sh
(p eaks
hk l )
and minor additio n
pm. T he calculated p eak positions are
show n as tick marks below each pattern. \G " denotes di˜raction
material.
alloy collected w ith
at 23.1 GPa corresp onds to a tw o- phase
peaks from the gasket
Phase T ransf or mat i ons i n Bi nar y A l l oys
681
.. .
W e ha ve p erform ed hi gh- pressure studi eson G a2 0 Sn8 0 al l oy, i ni ti al l y conta i ni ng Ù - Sn and G a. The pressure i ncrease wi tho ut heati ng results i n tra nsform ati ons
i n b oth com ponents: at pressures aro und 23 G Pa two pha ses were observed, bct- G a
(In typ e) and b ct- Sn, i n agreem ent wi th the data f or pure comp onents [23, 20] as
shown i n Fi g. 3. We ha ve exami ned thi s al loy af ter heati ng at 150 C for about
1.5 h under pressure 11 .4 G Pa. A di ˜ra cti on pattern ta ken at ro om tem p erature
at 11.4 G Pa after thi s heati ng reveal ed a new pha se wi th an sh structure to gether
wi th a sm al l am ount of bct- G a di ˜ra cti on p eaks. The l atti ce pa ram eters for the
sh pha se are a = 2 9 9 : 5 (2 ) pm , c = 2 7 7 : 9 (3 ) pm , and c= a = 0 : 9 2 8 at 11.4 G Pa.
The com po siti on of the sh pha se i s close to 15 { 18 at. % Ga . The sh pha se is sta bl e
i n the al l oy wi th pressure i ncreasing up to ¤ 3 0 G Pa reached i n thi s study and
decomp oses on decreasing pressure .
Thi s result conÙrm s the hom ol ogy rul e for bi nary al loy system s under pressure. The electro n concentra ti on of the sh pha se in G a{ Sn corresponds to a pa rti cul ar sta bi l i ty reg io n aro und 3.8 el./ ato m of the sh pha ses i n In{ Sn and Hg{ Sn. The
structura l data for sh i n G a{ Sn gi ves the val ue of the axi al ra ti o whi ch p erfectl y
agrees wi th the c= a = 0 : 9 3 , kno wn f or the mo st sh pha ses.
£
2. 3. T he In- based binar y al loys wi th C d, Sn and Pb
The gro up I I I element In crysta l l i zes i n a body-cent ered t etragonal (b ct) structure, tI 2 , space gro up I 4 = mmm , whi ch i s a sm al l di sto rti on of the face-center ed cubi c, fcc, structure. Theref ore, i t is m ore conveni ent to descri b e i t as a face-cent ered
t etragonal , fct, structure wi th an axi al ra ti o c= a = 1 : 0 7 5 [24]. Indi um rem ai ns i n
thi s structure under pressure at l east up to 50 G Pa [23], wi th slight vari ati on i n
c= a , the fcc pha se ha s no t been observed for In.
Ho wever a chang e fro m f ct to fcc occurs by al l oyi ng of In wi th gro up I I
m etal s Cd or Hg [24]. The addi ti on of these elements of l ower val ency decreases
c= a of the f ct structure resul ti ng i n the fcc structure a t ¤ 6 at. % of Cd or Hg . An
addi ti on of m etal s wi th hi gher val ency, as Sn or Pb, results in an i ncrease i n c= a
up to 1.10. Ho wever at concentra ti ons hi gher tha n ¤ 1 2 at. % Sn or Pb c= a j um ps
fro m c= a > 1 to c= a < 1 [24]. The existence of pha se regi ons fct c= a > 1 and fct
c= a < 1 i s shown i n the pha se di agra m of In{ Sn i n Fi g. 1. The vari atio n of the
axi al ra ti o on the al l oy com p ositi on for tetra gonal pha ses i n In al loys i s gi ven i n
Fi g. 4.
Our recent hi gh- pressure studi es [6] on Cd 1 0 In 9 0 al l oy show a f cc-to- fct tra nsiti on at pressure ¤ 1 : 5 GPa . The axi al ra ti o j um ps di sconti nuousl y fro m 1 to 1.04
and then it i ncreases wi th pressure up to 1.074 at 30 G Pa di splaying b ehavi our
sim i lar to pure In (see Fi g. 4).
Our further studi es [6] were p erform ed on the fcc pha se i n In{ Pb al l oys
exi sti ng under ambi ent condi ti ons at the Pb content hi g her tha n 30 at. % [24]. The
al l oys wi th 40 and 60 at. % Pb b oth show a tra nsi ti on fro m fcc to fct, whereas
c= a j um ps fro m 1 to c= a < 1 as shown i n Fi g. 4. W i th the pressure i ncrease the
682
V .F. Degtyareva
Fig. 4.
A xial ratio vs. electron concentrations
for fct phases in In- alloys at ambient
pressure (op en symb ols), based on results of Ref . [24] , and at high pressure (crossed
symb ols) from our studies. T he full circle and the cross corresp ond to the data on In at
normal and at pressure 23 GPa taken from Ref. [21] . T he arrow s show the change of
under pressure from the v alue j ust at the fcc{f ct transition
pressure reached ( ¤
vs.
n
30
GPa). T he full curves represent the calculated distortion
) w hen the Brillou in zone corners of typ e
sphere determined
by
n
W
and
W
0
( c =a
touch the free electron Fermi
. T he Brillou in zone of the fcc structure
right corner. F or exp erimental
c =a
to the value at the highest
is show n in the upp er
details see Ref . [6].
tetra gona l di sto rti o n i ncreases, a t the sam e pressure c= a i s l ower fo r the hi gher
Pb content. Thi s result was qui te puzzl ing unti l the com mon pl ot was dra wn for
tetra gona l pha ses i n In- based al l oys, Fi g. 4.
The studi es on In 8 0 Sn2 0 al l oy [6] wi th the i ni ti al pha se fct c= a < 1 al so show
an i ncrease in tetra gona l di sto rti on wi th the i ncrease i n pressure. The observed
chang es i n c= a for thi s al l oy were l ess pressure dep endent, howev er these correl ate
to the comm on trend i n the vari atio n of c= a al ong the electro n concentra ti on,
Fi g. 4. The dep endence of the axi al ra ti o on the m ean num ber of val ence electro ns,
n , indi cates an el ectro ni c ori gi n of the tetra gonal di sto rti on as wi l l be considered
b elow.
2.4. T he In{Bi
al l oy syst em
In the In{ Bi al l oy system [9,10 ] there are at am bi ent pressure three i nterm edi ate com p ounds In 2 Bi , In 5 Bi 3 , and InBi , Fi g. 5. Al l three com p ounds exi st
at certa i n sto i chiom etri c com p ositi ons and ha ve crysta l structures of site- ordered
typ e.
Phase T ransf or mat i ons i n Bi nar y A l l oys
Fig. 5.
.. .
683
C omparison of the phase sequences along the comp osition in the I n{Bi alloy
systems at the ambient
pressure (upp er plot),
based on data from [9], and at high
pressure (low er plot), based on the results from Ref s. [8, 25] .
The InBi com p ound was studi ed by energy- di spersive di ˜ra cti on wi th synchro tro n ra di ati on i n a di am ond- an vil cell under pressure up to 75 G Pa [3]. Ini ti al
tetra gona l structure is tP 4 , space gro up P 4 = nmm , wi th c= a = 0 : 9 5 . Fi rst tra nsform ati on occurs wi th a stro ng com pression al ong c to a simi l ar structure wi th
c= a = 0 : 6 5 and then to a tetra gona l b ody- centered structure, b ct, whi ch i s of a
ra ndo m (di sordered) typ e. The b ct pha se shows an i ncrease i n c= a fro m 0.91 to
0.96 i n the pressure ra ng e fro m 20 to ¤ 7 0 GPa . Som e evi dences was obta i ned for
a tra nsiti on to a cubi c b ody- centered structure, bcc. The b eha vi our of the I II{ V
comp ound InBi i s simi l ar to tha t of the i soelectro ni c gro up IV el ement Sn. The
i m p orta n t peculi ari ty of thi s sim i l ari t y i s the same upp er val ue of the axi al ra ti o
of the b ct structure atta i ned i n b oth cases.
The In 5 Bi 3 comp ound crysta l l i ses i n a tetra gonal structure wi th l atti ce para m eters a = 8 5 4 : 4 pm and c = 1 2 6 8 pm , wi th 32 atom s in the uni t cell , space
gro up I 4 = mcm [24]. R ecent studi es under pressure [8] shown tha t thi s structure
rem ai ned sta bl e at l east up to 25 G Pa. Ho wever, after annea l i ng of the In 5 Bi 3 sam pl e under pressure 15 G Pa at 15 0 C for 4 hours the di ˜ra cti on pa ttern chang es
p oi nti ng out a pha se tra nsiti on (see Fi g. 6).
No sol uti on for thi s pattern was found by assumi ng thi s sta te to b e a
sing le-pha se sta te and theref ore an attem pt to account for by a two- pha se mi xture
wa s made. A tenta ti ve sol uti on wa s found as a two -phase mi xture where b oth
pha ses a re b ody- centered tetra gona l, space gro up I 4 = mmm , di ˜eri ng by the axi al ra ti o. W e assign one pha se wi th the axi a l ra ti o c= a = 0 : 9 2 as b ct c= a < 1 ,
and the other pha se wi th c= a = 1 : 1 3 as b ct c= a > 1 . The ato m i c vol um es of
b oth these pha ses ha ve sli ghtl y di ˜erent val ues whi ch al l ow an esti mati on (o n the
gro und of V egard ' s l aw) the di ˜erence i n the pha se com p ositi ons as ab out 10 at. %.
Thi s observati on i m pl ies tha t the In 5 Bi 3 com p ound decom poses i nto two pha ses
of di ˜erent comp ositi on: one i s abo ve and one i s b elow of the i ni ti al In 5 Bi 3 com £
684
V .F. Degtyareva
Fig. 6.
A ngle disp ersive di˜racti on spectra (integrated
proÙles) from the In 5 Bi 3 com-
p ound (upp er) at 17. 8 GPa w ithout heating ; (low er) at 15 GPa af ter anneali ng at 150 C ,
£
4 h, corresp ondi ng to a decomp osition on a tw o- phase mixture
b ct
c =a >
1.
\G"
denotes peaks from the gasket material.
w ith bct
c =a
For exp erimental
<
1
and
details see
Ref . [8] .
p ositi on. Thi s tra nsf orm ati on i s reversi bl e, and the In 5 Bi 3 comp ound i s recovered
on decom pression.
In the Bi -ri ch regi on of the In- Bi al l oy system several i nterm edi ate pha ses
were obta i ned by \ pressure quenchi ng" [1], as a sim pl e hexagona l (sh) pha se and
an ortho rho m bica ll y di sto rted sh pha se. A com pl ex pha se was found near the
comp ositi on In 3 0 Bi 7 0 wi th the crysta l structure determ i ned recentl y as C mca -oC 1 6
[25], rel ated to hi gh-pressure fo rm s of Si and G e [18, 19]. The regi on of C mca -oC 1 6
pha se shoul d b e extended at certa i n P ; T -condi ti ons up to pure Bi , b ecause Bi IV
pha se i s assumed to b e sim i l ar to C mca -oC 1 6 structure [26, 25].
Under hi gh pressure a sequence of meta ll i c ra ndo m l y occupi ed pha ses can
b e outl i ned wi th qui te extended regi ons of sta bi l i t y as shown schemati cal ly i n the
l ower pl ot of Fi g. 5. The structure of the pha ses i n thi s sequence i s determ ined
by the facto r of the numb er of val ence electro ns as f or so- cal l ed Hum e{ Rothery
Phase T ransf or mat i ons i n Bi nar y A l l oys
.. .
685
pha ses. The In- ri ch regi on of the In{ Bi al l oy system di splays the fo ll owi ng structura l sequence under hi gh pressure:
fct
c= a > 1
!
f ct
c= a < 1 (=
b ct
c= a > 1 )
!
b ct
c= a < 1 :
Thi s sequence i s consi stent wi th tha t for In{ Sn al l oy system under am bient and
hi gh pressure shown i n Fi g. 1.
There i s much sim i l ari t y i n the hi gh- pressure pha ses of bi nary and hi gh- pressure form s of gro up IV elements. Thi s i mpli es these pha ses to be com m on
representa ti ves o f p ol yv al ent sp meta ls and al l oys, ra ther tha n pa rti cul ar chara cteri sti cs of certa i n elements or gro ups.
3 . Th e Bai n p at h f cc { b cc an d Br i l lo u in zo n e con cep t
T etra gona l di sto rti ons fro m fcc to b cc structure s represent a so- cal l ed Ba i n
pa th. Bo th structures can b e describ ed wi th the comm on cell , whether bo dyp
- centered tetra gona l, b ct, wi th c= a equal to 2 and 1, o r face-centered tetra gonal ,
p
fct, wi th c= a equal to 1 and 1 = 2 for fcc and b cc, respecti vely. To understa nd the
sta bi l i ty of the tetra gonal pha se agai nst the cubi c one, one shoul d consider two
contri buti ons to the crysta l energy: (1 ) the electro sta ti c Ewa l d term and (2) the
el ectro ni c ba nd- energy contri buti on due to Ferm i- sphere{ Bri l l oui n-zone i ntera cti on. The electro sta ti c energy cal culated f or a com mon tetra gona l structure along
the deform ati on fro m fcc to b cc has two mi ni ma corresp ondi ng to c= a = 1 (f cc) and
p
c= a = 1 = 2 (b cc) [27]. Thi s i ndi cates tha t Ewa l d term f avours the hi gh sym m etry
cubi c structures and the tetra gona l di storti on i s caused by other term s.
Attem pti ng to understa nd the sta bi l i ty of the tetra gona l pha ses i n In- al loys,
Svechkarev [28] has supp osed a l oweri ng i n the crysta l energy, when BZ corners
to uch the FS. The pl ot of c= a vs. electro n concentra ti on, n , cal cul ated for BZ
to uchi ng FS by corners of typ e W and W , i s presented in Fi g. 4 (sol i d curves).
A conta ct of al l W -typ e corners wi th FS results in fcc pha se, as i t app ears at
n = 2 : 9 4 , i.e. by al l oying o f In wi th ¤ 6 at. % Cd or Hg .
For the In al l oys wi th the el ectro n concentra ti on n < 3 : 1 2 , the axi al ra ti o
fol l ows the upp er bra nch of c= a vs. n , as shown i n Fi g. 4. For the In al loys wi th the
el ectro n concentra ti on n > 3 : 1 5 , the tetra gona l di sto rti on fol l ows the lo wer bra nch
wi th c= a < 1 . Thi s pha se exists at am bient pressure i n the ra nge fro m ¤ 1 5 to
¤
3 0 at. % Sn or Pb. Our observati on of the fcc-to -fct tra nsiti on on In al l oys wi th
40 and 60 at. % Pb i ndi cates tha t pressure causes an extensi on of the fct regi on
to wards Pb. The val ues of c= a i n thi s extended regi on fol l ow the Svechkarev pl ot
c= a vs. n , pro vi di ng a new evidence for thi s m odel of BZ{ FS Ùtti ng .
Acco rdi ng to thi s model , one can consider the sta bi l i ty of either fcc or fct as
determ i ned by a ba lance b etween electro sta ti c and el ectro ni c band energy term s.
The l atter contri buti on to the crysta l energy app ears to becom e m ore signi Ùcant
under pressure, sta bi l i sing the tetra gona l di sto rti on. Fol lowing a l i near extra po lati on of the exp eri menta l data to ward b cc, the l atter i s expected to b e sta bl e at
0
686
V .F. Degtyareva
p
el./ ato m, as the pl ot c= a vs. n crosses c= a = 1 = 2 at n = 4 : 2 . The b cc
structure i s real i sed under pressure i n the heavy gro up IV elements Sn and Pb
[20, 29]. Howev er, a tetra gonal di sto rti o n occurs i n Sn under pressure before the
tra nsiti on to bcc, wi th the axi al ra ti o rea ching c= a = 0 : 9 6 (0 .68 i n the fct setti ng )
[20]. Si mil ar tetra gona l di storti on of b cc ha s b een observed under pressure i n the
i soel ectro ni c comp ound InBi , wi th the sam e axi al rati o [3]. The dashed l i ne i n
Fi g. 7 extendi ng fro m thi s p oint at n = 4 towards l ower el ectro n concentra ti on
represents tetra gonal di sto rti ons f or b ct i n the Sn- ri ch al l oys wi th In and Hg under
pressure [4, 5, 7].
Thus the comm on pl ot of the Ba i n path fro m fcc to b cc as a functi on of c= a
vs. n represents a l i near b eha vi our of c= a i n the m i ddl e part wi th stro ng devi ati ons
fro m l i neari ty by appro achi ng b oth fcc and b cc.
R ecent observati on of decom positi on under pressure of the In 5 Bi 3 com pound
( n = 3 :7 5 ) and f orm ati on of two b ct pha ses wi th di ˜erent c= a [8] corresp onds to
the pro p osed mo del. Axi al ra ti os of b oth b ct pha ses fal l onto the pl ot j ust i n
the regi on of the c= a j um p fro m c= a > 1 to c= a < 1 for b ct pha ses, Fi g. 7.
n
¤
4
Fig. 7.
A representative
plot for axial ratios of the tetragonal
related alloys as a function of electron concentration.
to
c =a
phases for I n{Sn and
Dotted horizontal
lines corresp ond
v alues for fcc and bcc structures. T he solid lines represent average values of axial
ratios for I n alloys with C d, H g, Sn, Pb, and Bi at ambient pressure, whether stable [24]
or metastable [30]. T he dashed lines represent axial ratios observed at high pressure for
the alloys In{Sn, In{Pb
[6] (middle branch),
and Sn{H g, Sn{I n, I n{Bi (low er branch)
[3{5, 7, 8] . T he dot- dashed vertical line indica tes the comp osition of the initial
I n 5 Bi 3
comp ound, and horizontal
phases
w ith di˜erent
c =a
arrow s indicate its decomp ositi on into tw o tetragonal
ratios and di˜erent comp ositions, denoted by open triangles. T he solid
triangle denotes the high pressure bct phase in I nBi [3].
Phase T ransf or mat i ons i n Bi nar y A l l oys
.. .
687
4 . Co n cl u si o n
Hi gh pressure studi es on bi na ry al l oys show speciÙc tra nsf orm ati ons l i ke a
form ati on of new i nterm edi ate pha se fro m a two- pha se mi xture and i nversely | a
decomp ositi on of a ho mogeneous pha se i nto two pha ses of di ˜erent com positi on.
These tra nsform ati ons are al l owed for a two -com p onent system , due to an a ddi ti ona l vari abl e | com p ositi on, whi ch pro vi des for the pha se equi l i bri a one degree
of freedom m ore tha n for a one-comp onent system i n accorda nce wi th the G i bbs
pha se rul e. For i nsta nce, the sh pha se i n the In 2 0 Sn8 0 al l oy decom p oses to hcp and
b ct pha ses ab ove 10 G Pa and, i nv ersely, the sh pha se i s form ed under pressure
fro m the m i xture of Sn a nd G a i n the G a2 0 Sn8 0 al l oy. The In 5 Bi 3 comp ound was
found to decomp ose under pressure i nto two b ct pha ses wi th di ˜erent c= a and
di ˜erent com positi ons.
A bi na ry all oy system ha vi ng at ambi ent pressure a pha se di agra m wi th stoi chi ometri c com pounds or na rro w-ra ng e pha ses evol ves under pressure to a pha se
di agra m wi th a sequence of m etal l i c pha ses whi ch are mostl y of ra ndo m or substi tuti ona l typ e wi th qui te extended regi ons of stabi l ity . The com positi on ra ng es of
hi gh- pressure pha ses do no t rel ate di rectl y to tho se of l ow-pressure com pounds or
pha ses.
The structure and sta bi l i ty regi ons of the hi gh- pressure pha ses i n the bi na ry
sp m etal al l oys are determ i ned m ainl y by a facto r o f the el ectro n concentra ti on as
for the electro ni c Hum e{ Rothery pha ses.
Ac kn owl ed gm en t s
The autho r woul d li ke to tha nk Pro f. E. G. Po ny ato vski i , Pro f. W .B. Hol zapf el,
D r. F. Po rsch and O. D egtya reva for di scussions and col l ab orati on. Pa rti al supp ort
fro m the Russian Founda ti on for Ba sic R esearch by the gra nt No . 01- 02-97030 i s
gra tef ul l y ackno wledg ed.
R ef er en ces
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