PHYSICAL REVIEW B 94, 024102 (2016)
A alanche c i icali ies and elas ic and calo ime ic anomalies o he ansi ion om
cubic Cu-Al-Ni o a mix u e o 18Rand 2Hs uc u es
Edua d Vi es,1Jo di Ba ´
o,1Ma ´
ıa Ca men Galla do,2Jos´
e-Ma ´
ıa Ma ´
ın-Olalla,2F ancisco Ja ie Rome o,2Sa ah L. D i e ,3
Michael A. Ca pen e ,3Ekha d K. H. Salje,3Ma celo S ipcich,4,5Rica do Rome o,4and An oni Planes1
1Depa amen d’Es uc u a i Cons i uen s de la Ma `
e ia, Facul a de F´
ısica. Uni e si a de Ba celona,
Diagonal, 647, E-08028 Ba celona, Ca alonia, Spain
2Depa amen o de F´
ısica de la Ma e ia Condensada, Uni e sidad de Se illa, P.O. Box 1065, E-41080 Se illa, Spain
3Depa men o Ea h Sciences, Uni e si y o Camb idge, Downing S ee , Camb idge CB2 3EQ, Uni ed Kingdom
4IFIMAT, Uni e sidad del Cen o de la P o incia de Buenos Ai es and CICPBA, Pin o, 399, 7000 Tandil, A gen ina
5Consejo Nacional de In es igaciones Cien ´
ı icas y T´
ecnicas, A. Ri ada ia 1917, Buenos Ai es, A gen ina
(Recei ed 24 Feb ua y 2016; e ised manusc ip ecei ed 10 May 2016; published 5 July 2016)
We s udied he wo-s ep ma ensi ic ansi ion o a Cu-Al-Ni shape-memo y alloy by calo ime y, acous ic
emission (AE), and esonan ul asound spec oscopy (RUS) measu emen s. The ansi ion occu s unde cooling
om he cubic (β,Fm3m) pa en phase nea 242 K o a mix u e o o ho hombic 2Hand monoclinic 18R
phases. Hea ing leads i s o he back ans o ma ion o small 18Rdomains o βand/o 2Hnea 255 K, and
hen o he ans o ma ion 2H o βnea 280 K. The o al ans o ma ion en halpy is HT=328 ±10 J/mol and
is obse ed as one la ge la en hea peak unde cooling. The back- ans o ma ion en opy unde hea ing b eaks
down in o a la ge componen 18R o βa 255 K and a smalle , smea ed componen o he ans o ma ion 2H o
βnea 280 K. The p opo ions inside he phase mix u e depend on he he mal his o y o he sample. The elas ic
esponse o he sample is domina ed by la ge elas ic so ening du ing cooling. The weakening o he elas ic
shea modulus shows a peak a 242 K, which is ypical o he o ma ion o complex mic os uc u es. Cooling
he sample u he leads o addi ional changes o he mic os uc u e and domain wall eezing, which is seen by
g adual elas ic ha dening and inc easing damping o he RUS signal. Hea ing om 220 K o oom empe a u e
leads o elas ic anomalies due o he ini ial ans o ma ion, which is now shi ed o high empe a u es. The
ansi ion is smea ed o e a wide empe a u e in e al and shows s ong elas ic damping. The shea modulus o
he cubic phase is eco e ed a 280 K. The phase ans o ma ion leads o a alanches, which we e eco ded by AE
and by ime- esol ed calo ime y. The cooling ansi ion shows e y ex ended a alanche signals in calo ime y
wi h powe -law dis ibu ions. Cooling and hea ing uns show AE signals o e a la ge empe a u e in e al abo e
260 K. Spli ing he ans o ma ion in o wo ma ensi e phases leads o powe -law exponen s ε∼2(β↔18R)
and ε∼1.5(β↔2H) while he phase mix u e shows an e ec i e AE exponen o 1.7.
DOI: 10.1103/PhysRe B.94.024102
I. INTRODUCTION
Ma ensi ic ansi ions a e di usionless s uc u al ansi-
ions ha in ol e a change om high o low symme y
phases domina ed by a shea mechanism [1]. Usually, hese
ansi ions show a he mal cha ac e o some app oxima ion,
and hus p oceed ia a sequence o a alanches associa ed wi h
discon inui ies o he o de pa ame e ha e lec he ac ha ,
when ex e nally d i en, he sys em jumps ac oss a sequence o
me as able s a es. A alanches a e a consequence o dynamical
cons ain s imposed by nonhomogenei ies associa ed wi h
bo h in insic diso de such as la ice de ec s, impu i ies, o
by jamming, oge he wi h long- ange in e ac ion a ising om
elas ic compa ibili y cons ain s [2–4]. A alanches in many
such sys ems occu wi h he absence o cha ac e is ic ime
and size scales. This beha io de ines he so-called a alanche
c i icali y whe e he dynamics o he ans o ma ion p ocess is
cha ac e ized by a powe law o he a alanche size o ene gy
p obabili y dis ibu ion unc ion. Unde a he mal condi ions,
he powe -law exponen s ha e been a gued o depend on he
d i ing mechanism [5,6] and compu e simula ions ound no
shi s o he exponen s [7] when he mic os uc u al ans o -
ma ion p ocess emained opologically in a ian . This esul
sugges s ha exponen s may change as unc ion o a ian
mul iplici y, which is he numbe o equi alen s uc u al
domains ha can occu a he ansi ion [8]. This e lec s he
idea ha by inc easing a ian mul iplici y, he sys em is able o
ind mo e pa hs connec ing high- and low- empe a u e phases,
which may a ec exponen s o he a alanche size and ene gy.
Recen ly he e ec o in e nal winning o ma ensi ic a ian s
on p opaga ion dynamics has also been sugges ed o in luence
a alanche dynamics du ing s ess-induced ans o ma ion o
an o ho hombic ma ensi e in Cu-Al-Ni [9]. To u he cla i y
he in luence o symme y e ec s on a alanche c i icali y,
we s udy a alanches in he he mally induced ma ensi ic
ansi ion o a Cu-Al-Ni alloy, which ans o ms o a mix u e
o monoclinic and o ho hombic phases in ol ing a ian
mul iplici ies o 12 and 6. We show ha he powe -law
exponen s do indeed change o di e en hea ing and cooling
p o ocols due o his o y-dependen e ec s on he selec ed
ansi ion pa hs.
We moni o ma ensi ic a alanches om acous ic emission
(AE) and calo ime ic measu emen s. AE o igina es om as
local changes o s ain ields ac oss p opaga ing in e aces.
I is a e y sensi i e echnique ha allows he de ec ion
o weak e en s o e a wide ange o leng h scales om
nanome e s o mic ons. Acous ic AE wa es ca y empo al
and spa ial in o ma ion ela ed o he sou ce o he acous ic
emission, and ha e been used ex ensi ely in he pas o s udy
2469-9950/2016/94(2)/024102(8) 024102-1 ©2016 Ame ican Physical Socie y
EDUARD VIVES e al. PHYSICAL REVIEW B 94, 024102 (2016)
ma ensi ic phase ansi ions (see, o ins ance, Re . [10,11]).
Recen ly, by modeling AE wa e o ms o he acous ic e en s
i has been shown ha he ans o ma ion om cubic o
o ho hombic in Cu-Al-Ni occu s on a as e imescale han
he ans o ma ion o he monoclinic phase [12]. He e we
ocus on ano he aspec o he ansi ion mechanism, namely
he collec i e dynamics o he ansi ion om he iew o
s a is ical mechanics. We measu e he ene gy and ampli ude
o AE e en s du ing he ans o ma ion p ocesses [8] and
combined hese measu emen s wi h calo ime ic and RUS
measu emen s, which allows us o dis inguish be ween wo
di e en ans o ma ion mechanisms.
Cu-Al-Ni belongs o he amily o Cu-based Hume-Ro he y
shape-memo y alloys. In his class o alloys, as cooling
om he high- empe a u e (diso de ed bcc phase) s abili y
egion p e en s he o ma ion o equilib ium phases and hus
enables e aining o an o de ed (nea es - and nex -nea es -
neighbo o de ing) cubic phase (usually deno ed βphase)
a (o close o) oom empe a u e. A e u he cooling,
his long-li ing me as able phase unde goes a ma ensi ic
ansi ion. This ma e ial has been he subjec o ac i e esea ch
du ing ecen yea s, since associa ed wi h he ma ensi ic
ansi ion, i displays in e es ing shape memo y and su-
pe elas ic p ope ies [13]. P oblems such as he in luence
o aging [14,15] and cycling [16] on he ansi ion and
ma ensi ic mic os uc u e, and he elas ic beha io abo e and
below he ans o ma ion [17,18] among o he s, ha e been
s udied in de ail due o hei ele ance in ela ion o po en ial
echnological applica ions o his ma e ial.
In Cu-based alloys, phase s abili y is la gely con olled by
he elec on concen a ion, e/a [19]. In pa icula , he ac ual
ma ensi ic s uc u e is sensi i e o ine uning o he elec on
concen a ion e/a [20]. In Cu-Al-Ni he s abili y bounda y
sepa a ing he monoclinic 18R om he o ho hombic 2H
phase has been p edic ed a e/a ∼1.53 [21]. I has been
epo ed ha bo h monoclinic and o ho hombic phases can
coexis [22–24] close o his c i ical elec on concen a ion.
The exis ence o his coexis ence egion is also a o ed by
he ac ha he s esses associa ed wi h he occu ence o
o ho hombic a ian s a he onse o he empe a u e-d i en
ansi ion may induce he ans o ma ion owa ds he mono-
clinic phase [25]. The mally induced ansi ions equally occu
so ha he me hodology o he in es iga ion o e oelas ic
phase ansi ions becomes applicable [26] while he a he mal
na u e o he phases emains app oxima ely p ese ed o e
limi ed empe a u e in e als [27].
II. EXPERIMENT
A Cu-Al-Ni polyc ys alline sample was p epa ed by al-
loying 99.99% pu i y coppe , aluminum, and nickel. Small
pieces wi h app op ia e masses (weigh ed wi h a p ecision o
0.1 mg) o each me al we e cu and we e mel ed by means o
a W-2%Th elec ode acuum a c u nace in o a wa e -cooled
coppe c ucible unde a pa ial a gon a mosphe e. The sample
was emel ed a leas eigh imes in o de o each a high
homogenei y deg ee. A e solidi ica ion, he Cu-Al-Ni alloy
was homogenized a 1073 K o abou wo days. I was hen
quenched in wa e and annealed a oom empe a u e o
some weeks. This hea ea men ensu es a highly o de ed
FIG. 1. G ain size dis ibu ion o he s udied sample. The con in-
uous line is a log-no mal i .
s a e, ee om in e nal s esses and wi h minimum acancy
concen a ion. This hea ea men esul ed in a la ge g ain
size wi h a e age diame e o abou 1 mm (see Fig. 1).
The composi ion was ob ained om EDX measu emen s o
be Cu68.1Al28.1Ni3.8(e/a =1.56). A high empe a u e, he
sample displays an L21(Fm3m) cubic s uc u e (βphase).
Fo calo ime ic and AE measu emen s he same specimen was
used. I was cu om he ingo wi h a low-speed diamond saw.
The specimen is 2.4 mm all and has a la nea ly ec angula
base o 53.5 mm2. The co esponding mass is 0.9921 g. Fo
RUS measu emen s we cu a slice o 4 mm ×2.4 mm ×1 mm,
pa allel om one side o he specimen used o calo ime y and
AE expe imen s.
The he e ogeneous s ains gene a ed by he polyc ys alline
na u e o he sample we e analyzed by x- ay di ac ion using
a William-Hall analysis. The esul ing William-Hall plo s
con i m la ge g ain sizes and mode a e s ain b oadening. The
he e ogeneous s ain along he cubic main axes {100}is 2.4%
and 0.5% along {110}di ec ions. This means ha g ains a e
squeezed along he {100}di ec ions bu a he uns essed along
he {110}di ec ions. This is ela ed o he di ec ion o he shea
mechanism leading o he ma ensi ic s uc u e om he pa en
cubic phase [20]. The o e all s ain e ec is, hus, a he small
and no unusual o ma ensi ic ans o ma ions.
Measu emen s o hea lux φwe e pe o med using a high-
esolu ion conduc ion calo ime e , which has been desc ibed
elsewhe e [28]. The sample is p essed be ween wo iden ical
hea lux me e s, which a e made om 50 ch omel-cons an an
he mocouples connec ed in se ies, wi h he wi es placed in
pa allel lines. The lux me e s a e he mally coupled o a
la ge calo ime ic block. The block is placed in o a he me ic
ou e case unde acuum (10−7 o ). The empe a u e o
he calo ime ic block is eco ded by means o a pla inum
esis ance he mome e .
Hea luxes we e measu ed wi h a esolu ion <0.1μW.
Due o he high he mal ine ia o he calo ime e block, i was
possible o pe o m expe imen s a e y low- empe a u e a es,
, in he ange 10−4–10−5K/s. Tempe a u e luc ua ions o he
calo ime e block we e smalle han 10−6K. The sys em wo ks
as di e en ial he mal analysis (DTA) de ice du ing cooling
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AVALANCHE CRITICALITIES AND ELASTIC AND . . . PHYSICAL REVIEW B 94, 024102 (2016)
and hea ing uns. The elec omo i e o ce p o ided by he lux
me e s, which is p opo ional o he hea lux, was eco ded by
a nano ol me e Kei hley K2182 a a sampling a e o 12.5 Hz.
The in eg al o he lux abo e a sui able baseline is p opo ional
o he excess en halpy o he sample [29].
The expe imen al a angemen o AE measu emen s is
based on a PCI-2 acquisi ion sys em (Eu ophysical Acous ics),
wo king a a nominal ime esolu ion o 40 MHz. The sample
was moun ed on he op o a coppe block si ua ed inside a
double Fa aday cage cons i u ed o coppe and i on shielding
laye s. A piezoelec ic senso (R15LT ansduce , encapsula ed
in s ainless s eel) was acous ically coupled o he uppe su ace
o he sample by a hin laye o pe oleum jelly. The signal
om he ansduce was ampli ied (60 dB), band il e ed
be ween 200 kHz and 1 MHz, and ans e ed o he acquisi ion
sys em. Fo iden i ica ion o AE e en s, a h eshold abo e he
(una oidable) ins umen al noise was ixed a 24 dB. An e en
iwas assumed o s a wi h he i s c ossing o he h eshold
a ime i. The end o he e en , i+ i, is de e mined when
he ol age c osses he h eshold in he downwa d di ec ion
and emains below h eshold o mo e han a p ese de ec ion
ime o 100 μs in ou expe imen s. The ene gy o he e en s
is de e mined by nume ical in eg a ion o he squa e ol age
du ing he du a ion  i, no malized by a e e ence elec ic
esis ance o 10 k. The AE ac i i y is de ined as he numbe
o e en s pe uni ime measu ed o e la ge enough in e als (o
he o de o he second) so ha his quan i y can be compa ed
wi h mo e mac oscopic measu emen s such as calo ime y. A
co esponding AE ene gy can also be de ined as he sum o
he ene gies o he e en s emi ed o e hese la ge in e als.
The p inciples o RUS ha e been desc ibed in de ail by
Miglio i and Sa ao [30] and Miglio i and Mayna d [31].
RUS is used o measu e he esonan equencies o no mal
modes o ib a ion o a ma e ial in he equency egion
∼0.1–2 MHz. In he low- empe a u e RUS a angemen used
o he expe imen s desc ibed he e, he sample sa ligh ly
be ween piezoelec ic ansduce s in a sample holde , which
was lowe ed in o an o ange helium low c yos a wi h a ew
mba s o helium as exchange gas [32]. Acous ic esonances
o he sample we e exci ed by sending an ac elec ic signal
o one o he ansduce s, and we e de ec ed by he second
ansduce . Indi idual spec a, consis ing o 130 000 da a
poin s, we e collec ed in cooling and hea ing sequences,
including a he mal equilib a ion pe iod o 20 min a each
empe a u e. Spec a we e analyzed o line using he so wa e
package IGOR (Wa eme ics) o i selec ed peaks wi h an
asymme ic Lo en zian unc ion. The squa e o he esonance
equency, , scales wi h he elas ic cons an s which de e mine
a gi en mode and hese, in u n, depend p ima ily on shea ing
mo ions. The peak wid h a hal heigh ,  , p o ides a measu e
o acous ic loss h ough he in e se mechanical quali y ac o ,
Q−1= / .
III. RESULTS
DTA aces, de ined as he a io, φ/ , o hea lux and
empe a u e a e, we e ob ained o hea ing and cooling uns.
Two alues o he empe a u e a e we e used, =0.04 K/h
(∼10−5K/s) and =0.25 K/h(∼7×10−5K/s). Typical
esul s a e shown in Fig. 2. The di ec ans o ma ion (cooling
2
3
4
5
6
7
8
9
10
230 240 250 260 270 280 290
φ/ (J/K)
Tempe a u e
(
K
)
FIG. 2. DTA aces o hea ing ( ed) and cooling (blue) expe i-
men s. Da a co esponding o cooling expe imen s ha e been changed
in sign o allow a be e compa ison o hea ing da a.
un) is cha ac e ized by a b oad anomaly o he hea lux
be ween 250 K and 230 K wi h a la ge numbe o spikes o e
a smalle empe a u e in e al a he uppe end o he b oad
anomaly. The e e se ans o ma ion (hea ing da a) shows a
smea ed anomaly be ween 235 K and 260 K whe e a small
numbe o weak spikes a e supe imposed, ollowed by a la ge
numbe o s ong hea lux peaks in he empe a u e in e al
be ween 265 K and 295 K. This sugges s ha on hea ing he
ansi ion is spli in o wo di e en s ages.
In eg a ion o he DTA ace gi es he ansi ion en halpy
(Fig. 3), HT=328 ±10 J/mol. The low- empe a u e ans-
o ma ion (smoo h anomaly wi h some spikes) accoun s o
94% o o al en halpy. The high- empe a u e ans o ma ion
accoun s o he emaining 6%. The ac ion o en halpy as
unc ion o he empe a u e H(T)/HTis aken as measu e
o he ans o med ac ion x(T).
-350
-300
-250
-200
-150
-100
-50
0
230 240 250 260 270 280 290
0.00
0.20
0.40
0.60
0.80
1.00
ΔH (J/mol)
T ans o med ac ion x
Tempe a u e
(
K
)
FIG. 3. In eg a ed ansi ion en halpy o hea ing ( ed) and
cooling (blue) uns. The o al en halpy change is iden ical wi hin
expe imen al e o s. The igh scale gi es he co esponding ans-
o med ac ion.
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0
100
200
300
220 240 260 280 300
Tempe a u e (K)
(b) Cooling
0
50
100
150 (a) Hea ing
Ac i i y (K-1) (E > 23 μJ)
FIG. 4. Ac i i y plo s o a alanches ob ained om calo ime ic
measu emen s du ing (a) hea ing and (b) cooling uns. Ac i i y is
measu ed by coun ing he numbe o spikes la ge han 23 μJin
in e als o hal a Kel in. Smoo h con inuous lines display a i o he
sum o wo Gaussian dis ibu ions.
High- equency noise in he calo ime ic da a was il e ed
ou , a e he expe imen , by a i h-o de all-pole Bu e wo h
il e wi h a no malized cu o equency o 8 ×10−2Hz.
The spikes o he expe imen s we e ob ained by sea ching he
local maxima in he hea lux signal. The numbe o spikes
la ge han 23 μJ is 929 o he cooling un and 1518 o he
hea ing un. The a alanche ac i i y was measu ed as numbe
o spikes pe in e al o 0.5 K. The empe a u e-dependen
ac i i y is shown in Fig. 4. The ac i i y shows a single
peak du ing he cooling expe imen , wo peaks a e obse ed
du ing he hea ing un. The en halpy change occu s mainly
du ing he low- empe a u e ans o ma ion while he spike
ac i i y is s onge a he high- empe a u e ans o ma ion.
The a alanches we e s a is ically analyzed and he exponen
o he powe -law p obabili y p(E)∼E−εwas de e mined
(Fig. 5, op panel) using a maximum-likelihood analysis. An
exponen ε=2.0±0.3 was ob ained o he cooling un. Fo
he hea ing un, he signals o he wo ans o ma ions we e
analyzed sepa a ely. An exponen ε=2.4±0.4 was ob ained
o he low- empe a u e ans o ma ion and ε=2.2±0.3was
ob ained o he high- empe a u e ans o ma ion, albei wi h
e y ew peaks in he he mal cu e.
In Cu-based shape-memo y alloys an en opy change o ca.
1.3 J/K mol was p edic ed o sys ems ans o ming o he
18Rphase and an en opy change o 1.6 J/K mol o sys ems
ans o ming o he 2Hphase [20]. Taking he ac ions o
bo h phases es ima ed om calo ime ic measu emen s and
100
101
102
103
10-4 10-3
Coun s
Ene gy (J)
Hea ing (T<260K)
Hea ing (T>260K)
Cooling
2.4
2.2
2.0
FIG. 5. Log-log plo o he dis ibu ion o ene gies (linea bins) o
he spikes de ec ed om calo ime ic measu emen s. Da a eco ded
du ing he hea ing un a e spli a T=260 K sepa a ing he
con ibu ions o he ac i i y obse ed in Fig. 4. S aigh lines display
bes powe -law i s o expe imen al da a.
a common equilib ium empe a u e T0≃250 K o bo h
β↔18Rand β↔2H ansi ions, we ob ain ha he hea
exchanged a he ans o ma ion should be, q≃1.3(J/mol
K) ×0.94 ×250 (K) +1.6(J/mol K) ×0.06 ×250 (K) =
329.5J/mol, which is e y close o he calo ime ically mea-
su ed ans o ma ion en halpy o 328 J/mol du ing he cooling
un. This esul is hence consis en wi h he in e p e a ion ha ,
on cooling, he ansi ion occu s o a mix u e o 94% 18Rand
6% 2H. The spli ing o he e e se ansi ions unde hea ing
is a consequence o he ac ha he ansi ion o he 2Hphase
is known o occu wi h a much wide he mal hys e esis han
he ansi ion o he 18Rphase [25,33].
An addi ional calo ime ic expe imen was pe o med o
s udy he he mal hys e esis o he wo ans o ma ions.
Fi s , he sample was cooled down o 200 K om oom
empe a u e (s ep 1); i was hea ed up o 260K (s ep 2) when
he low- empe a u e ansi ion appea ed inished. The sample
was hen ecooled o 220 K (s ep 3) and hea ed again o oom
empe a u e (s ep 4). DTA aces o his expe imen a e shown
in Fig. 6.
The da a o s ep 1 and s ep 2 a e simila o hose ob ained
in he ini ial expe imen s (see Fig. 3). Following he ideas
p e iously desc ibed [22,23], a mix u e o monoclinic 18Rand
o ho hombic 2Hg ows du ing he cooling ans o ma ion.
The ansi ion om monoclinic 18R o cubic βphase has been
comple ed a e s ep 2 and he sample con ains a mix u e o
he cubic and o ho hombic phases.
When he sample is cooled again (s ep 3), a b oad anomaly
is ound. The new ansi ion s a s a a highe empe a u e (by
app oxima ely 5 K) while he peak empe a u e o he hea
lux does no change. Ve y ew spikes occu in his expe imen
(a ound 20 e en s), which sugges s ha a alanches du ing he
cooling ans o ma ion a e due o he now-deple ed cubic o
o ho hombic ans o ma ion.
Finally, he sample is hea ed om 220 K o oom empe a-
u e (s ep 4). Two sligh ly o e lapped s ages we e ound. The
low- empe a u e ans o ma ion is smea ed wi h no spikes.
The high- empe a u e ans o ma ion shows a la ge numbe
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AVALANCHE CRITICALITIES AND ELASTIC AND . . . PHYSICAL REVIEW B 94, 024102 (2016)
2
3
4
5
6
7
8
9
10
230 240 250 260 270 280 290
φ/ (J/K)
Tempe a u e
(
K
)
S ep 1: cooling om oom emp. o 220K
S ep 2: hea ing om 220K o 260K
S ep 3: cooling om 260K o 220K
S ep 4: hea ing om 220K o oom emp.
FIG. 6. Hea lux e sus empe a u e o se e al uns: s ep1,
cooling om oom empe a u e o 220 K (blue line); s ep 2, hea ing
om 220 K o 260 K ( ed line); s ep 3, cooling om 260 K o 220 K
(black line); s ep 4, hea ing om 220 K o oom empe a u e (magen a
line).
o spikes (3712) which a e supe imposed o a b oad smoo h
anomaly. This smea ed anomaly was no p e iously obse ed
in he i s expe imen . The o al en halpy balance is ul illed
as shown i espec i e o he he mal his o y o he sample
(Fig. 7.)
We eco ded AE signals du ing hea ing and cooling wi h
a es 5.5×10−3K/s. AE a es a e usually much as e han
calo ime ic a es, which a e de e mined by he e y slow
esponse o he calo ime e . We s a ed measu emen s a a high
enough empe a u e o 320 K ha ensu es ha he sample is
ully in he βphase. The sample was hen cooled o 220 K and
subsequen ly hea ed om 220 K o 320 K. Figu e 8( op panel,
blue cu e) shows ha du ing cooling signi ican AE ac i i y
was de ec ed be ween 260 K and 230 K, wi h a maximum nea
240 K. The AE ac i i y peak is loca ed in he same empe a u e
in e al whe e a alanches in he hea lux we e ound. Also in
-350
-300
-250
-200
-150
-100
-50
0
230 240 250 260 270 280 290
0.00
0.20
0.40
0.60
0.80
1.00
ΔH (J/mol)
T ans o med ac ion x
Tempe a u e
(
K
)
S ep 1
S ep 2
S ep 3
S ep 4
Ini ial hea ing
FIG. 7. En halpy excess (ob ained om in eg a ion o calo ime -
ic cu es) e sus empe a u e o se e al uns: s ep 1, cooling om
oom empe a u e o 220 K (blue line); s ep 2, hea ing om 220 K o
260 K ( ead line); s ep 3, cooling om 260 K o 220 K (black line);
hea ing om 220 K o oom empe a u e (magen a line). The do ed
ed line ep esen s he en halpy o he ini ial hea ing un.
0
100
200
300
400
500
220 240 260 280 300 320
AE ac i i y (s-1)
Tempe a u e (K)
(b) Cooling
Hea ing
0
2000
4000
6000
8000
10000
220 240 260 280 300 320
AE ene gy (aJ)
(a)
FIG. 8. (a) AE ene gy cumula ed in bins o 2 s as a unc ion o
empe a u e du ing cooling om 320 K down o 220 K (blue) and
subsequen hea ing up o 320 K ( ed). (b) AE ac i i y eco ded du ing
he same uns (using also bins o 2 s). The a ows indica e he ange
o he analysis p esen ed in Fig. 9(b).
ag eemen wi h calo ime y, he ac i i y peak shows big spikes
a he ea ly s ages o he ans o ma ion and becomes smoo he
a he la e s ages. In con as , du ing hea ing AE appea s in a
e y b oad in e al be ween 240 K and 300 K (see Fig. 8, op
panel ed cu e). Two peaks we e iden i ied in his in e al by
calo ime y, while only one b oad peak is seen in AE. When
he AE ene gy is examined, ins ead o he AE ac i i y, (see
Fig. 8, op panel), a la ge amoun o he ene gy dissipa ed by
AE is associa ed wi h he big AE spikes while li le ene gy is
dissipa ed in he smoo he ans o ma ions egions.
We show he dis ibu ion o ene gies o one comple e se
o AE signals du ing cooling and hea ing uns as log-log
plo in Fig. 9(a). An app oxima e powe -law dis ibu ion wi h
100
101
102
103
104
105
0 20 40 60 80
Coun s
log10(E) (E in aJ)
(a)
Cooling
Hea ing
0 20 40 60 80 100
log10(E) (E in aJ)
(b)
Hea ing (T<247.7 K)
Hea ing (T>272.7 K)
FIG. 9. (a) Log-log plo o he dis ibu ion o a alanche ene gies
(in aJ) o comple e cooling and hea ing uns. (b) Dis ibu ion o
a alanche ene gies (in aJ) o signals de ec ed in he empe a u e
in e al 230–250 K (co esponding o he 18R→β ansi ion) and
270–290 K (co esponding o he 2H→β ansi ion) espec i ely.
The in e als a e indica ed in Fig. 8.
024102-5

EDUARD VIVES e al. PHYSICAL REVIEW B 94, 024102 (2016)
1
1.5
2
2.5
3
3.5
4
10-1 100 101 102 103 104 105 106
Exponen ε
E
min (aJ)
Cooling
Hea ing
Hea ing (T<247.7)
Hea ing (T>272.7)
1.66
FIG. 10. Likelihood plo s co esponding o he dis ibu ions
shown in Figs. 8(a) and 8(b).
some exponen ial cu o s is ob ained o e mo e han h ee
decades. The co esponding cha ac e is ic exponen s ha e
been ob ained using a maximum likelihood me hod, which
consis s o s udying he beha io o he i ed exponen as
unc ion o a a ying lowe cu o Emin (see, o ins ance,
Re . [34]). This analysis should lead o a pla eau ha de ines
he cha ac e is ic exponen . This is shown in Fig. 10.The
pla eau is no well de ined bu some egion wi h small slopes
appea s. The exponen was es ima ed a he onse o he pla eau
as indica ed by dashed lines in he likelihood cu es in Fig. 10.
The exponen s ob ained a e ε=1.7±0.2 o cooling and
hea ing uns. We ha e u he analyzed sepa a ely signals
co esponding o he ea ly 18R→βand la e 2H→βs ages
du ing hea ing. The co esponding ene gy dis ibu ions a e
shown in Fig. 9(b) and he esul s o he maximum-likelihood
analysis shown in Fig. 10. An ene gy exponen ε=2.0±0.15
(g een cu e) is ob ained o signals a he ea ly s ages while
an exponen ε=1.5±0.15 (o ange cu e) is es ima ed o
la e s ages. These alues a e in excellen ag eemen wi h
hose p edic ed o β↔18Rand β↔2H ansi ions,
espec i ely [8].
The esul s o he RUS measu emen s a e shown in Fig. 11.
Cooling he sample leads o a dec ease o he shea s i ness
o he sample. Figu e 11 shows da a om wo elas ic RUS
esonances. Bo h esonances show a weak bu signi ican
dec ease o he shea modulus abo e 260 K and a s ong
dependence on he he mal his o y o he sample. Fi s , cooling
om oom empe a u e o 220 K gi es a well-de ined ansi ion
a 242 K wi h signi ican so ening below 280 K. The phase
below 242 K is no s a ic and shows s ong he mal ha dening
unde cooling. Subsequen hea ing om 220 K o 260 K
leads o a much wide ansi ion a 249 K. A 260 K he
shea modulus has no eco e ed he s i ness o he i gin
c ys al du ing cooling. Repea ed cooling om 260 K o 220 K
eco e s he ansi ion a 242 K o he i s cooling expe imen .
The da a in he low- empe a u e phase a e ully ep oduced and
show no he mal hys e esis. Second hea ing om 220 K o
oom empe a u e leads o elas ic so ening, which is u he
shi ed o highe empe a u es. The ansi ion egion is now
e y b oad and he maximum damping and minimum shea
modulus is a ound 254 K, which is well abo e he ini ial
FIG. 11. RUS equencies (a) and damping (b) du ing cooling
and hea ing expe imen s. The sample was i s cooled o 220 K,
hen hea ed o 260 K, ecooled o 220 K and inally hea ed o
oom empe a u e. The minima in he squa ed RUS equency and
he maximum o he damping Q−1a e close o he ans o ma ion
ins abili y poin s, which show wi h he mal his o y. The g adual
changes below 242 K show ha he 18R+2Hphase mix u e is no
a he mal a low empe a u es.
ansi ion empe a u e o 248 K. The ini ial shea moduli a e
ully eco e ed abo e 280 K.
IV. DISCUSSION
The h ee expe imen al echniques disclose mechanisms
o he he mal ans o ma ion o a Cu-Al-Ni shape-memo y
alloy on di e en leng h scales. Calo ime y measu es he o al
ans o ma ion en halpy and, addi ionally, de ec s a alanches
as indica ion ha he ans o ma ion con ains non-smoo h
dynamics. This does no mean ha he en i e p ocess is
nonsmoo h as much o he en halpy is con ained in a
con inuous backg ound signal. In con as , AE is a e y
sensi i e echnique o small changes o he local s ain ield,
and hus e y adequa e o quan i y a alanche dynamics. RUS
measu emen s a e age o e imes ha a e much longe han he
li e imes o a alanches [35]. RUS measu es he o e all elas ic
shea moduli o he sample, which includes bo h in insic
so ening/s i ening due o coupling o he o de pa ame e
wi h s ain and ex insic con ibu ions domina ed by he
o ma ion and change o win and in e acial mic os uc u es.
Fi s cooling o Cu-Al-Ni leads o a main ans o ma ion
e en nea 242 K, which is obse ed by all echniques.
Calo ime y e eals ha he esul ing low- empe a u e o m
024102-6
AVALANCHE CRITICALITIES AND ELASTIC AND . . . PHYSICAL REVIEW B 94, 024102 (2016)
is a phase mix u e o 18Rand 2H. The p opo ions be ween
hese phases is s ongly biased owa d he 18Rphase, i.e.,
94%. In con as , he AE signal is biased owa ds he
ans o ma ion in o he 2Hphase. The RUS spec a show
an o e all elas ic so ening, which eaches a maximum a
he ansi ion poin . The de ails o he so ening a e unusual
o ma ensi es, howe e , whe e he shea modulus emains
empe a u e independen once he ma ensi e is o med. This
a che ypal beha io was epo ed, e.g., o Cu74.08Al23.13Be2.79
[36]. The mic os uc u e o he ma ensi e phases in Cu-Al-
Ni s ill changes a low empe a u es so ha he a he mal
beha io is ne e es ablished o hese phase ansi ions. This
obse a ion quali ies he e m a he mal: a 242 K he βphase
ans o ms in o 18Rand 2Hin a na ow empe a u e in e al
while he mic os uc u e o he phase mix u e changes unde
u he cooling. Du ing his he mal adap a ion [37]noAE
signals o je ky hea luxes a e ound. This means ha he
p opo ions be ween he 2Hand 18Rphases do no change in
any measu able way bu ha he win mic os uc u es do. The
he mal ha dening o he shea moduli is ully ep oducible
on cooling bu no on hea ing whe e we also ind a signi ican
he mal hys e esis. This indica es ha he elas ic ha dening
and inc ease o he elas ic damping (as exp essed by Q−1)
du ing cooling ela es o g adual domain wall eezing wi h an
ex emely b oad exci a ion spec um. Such he mal beha io
is commonly obse ed in e oelas ic ma e ials [38,39].
AE is expec ed o be associa ed wi h he amoun o
ans o ma ion hys e esis [40] so ha he AE esul s can be
unde s ood i a ans o ma ion o wo ma ensi ic phases 18R
and 2Hoccu s simul aneously du ing cooling. Du ing hea ing
hese wo ma ensi es e ans o m di e en ly. A he ea ly
s ages, he 18Rphase e ans o ms o he pa en phase (and
possibly some 2H) wi h a weak hys e esis and hus wi h AE
ca ying e y li le dissipa ed ene gy. A highe empe a u e,
he 2Hphase e ans o ms unde he emission o e y in ense
AE signals. The main AE ene gy nea 280 K is hen associa ed
wi h he ans o ming be ween he small amoun o 2Hand β.
We now e u n o he discussion o ene gy exponen s.
The pu e phase ansi ions β↔2Hand β↔18Rshow
ela i ely good powe -law dis ibu ions o a alanche ene gies
in bo h hea lux and AE measu emen s. The exponen s o
he indi idual ans o ma ions we e iden i ied unde hea ing.
AE measu emen s ende clea ly di e en exponen s, namely
ε=2(β↔18R) and ε=1.5(β↔2H). The exponen o
he majo i y ans o ma ion β↔18Ris a emo ed om
he expec ed mean ield alue[4] while he lowe alue may
coincide wi h mean- ield heo y i he bounda y condi ions
a e ca e ully chosen [41]. The phase mix u e shows an
in e media e exponen ∼1.7 in AE. The e o s in he exponen s
de e mined om calo ime ic measu emen s a e la ge and
disc imina ion o di e en exponen s is less clea . In ac , hea
lux measu emen s ende in gene al sligh ly la ge exponen s
han AE measu emen s. In ou case he disc epancies be ween
calo ime ic and AE de e mina ion o he exponen s seem o
be a consequence o a ela i ely bad s a is ics in he case o
calo ime ic measu emen s. I is wo h no ing ha he numbe
o de ec ed signals is a leas one o de o magni ude la ge
in he case o AE measu emen s. Ne e heless, we should no
igno e he ac ha AE and calo ime ic echniques enable
a alanche de ec ion in e y di e en ene gy in e als. The ac
ha he ob ained exponen s a e so simila ein o ces he idea
o scale in a iance o he a alanche p ocess.
Anin e es ingissue obediscussedinsomemo ede ailis
he di e en ene gy exponen s es ima ed o he β↔18Rand
β↔2H ansi ions. I has been sugges ed ha hese exponen s
depend essen ially on a ian mul iplici y, which is gi en by
he numbe o equi alen s uc u al domains ha can occu
a he ansi ions. Tha is, on he a io be ween he symme y
ope a ions in he pa en and ma ensi ic phases. F om his
pe spec i e, sys ems wi h di e en a ian mul iplici y should
belong o di e en uni e sali y classes. The idea behind his
poin o iew is ha by inc easing a ian mul iplici y, he
sys em is able o ind mo e pa hs connec ing high and low
symme y phases. Thus, as a ian mul iplici y inc eases he
p obabili y o la ge e en s should dec ease compa ed wi h he
p obabili y o small ones, which should gi e ise o a la ge
c i ical exponen . The a ian mul iplici y is 6 o he β→
2Hand 12 o he β→18R ansi ion. This coincides wi h a
change o ene gy exponen s om ε=1.5(β↔2H) oε=2
(β↔18R) so ha an inc ease in he mul iplici y also leads o
an inc ease o he exponen .
The inc ease o mul iplici y has ano he e ec ; i can
inc ease he smoo hness o he ansi ion. I he geome ical
ans o ma ion occu s using a g ea e numbe o in e media e
s epping s ones i can occu wi h ewe o no a alanches on
a local s age. In Cu-Al-Ni we ind ha he majo i y β↔
18R ans o ma ion is much smoo he han he β↔2H
ans o ma ion al hough he la e in ol es only a e y small
ac ion o he sample.
A simple model has ecen ly been p oposed based on
b anching andom p ocesses o deal wi h he change o
c i ical exponen wi h a ian mul iplici y [42]. The model
assumes ha he ma ensi ic mic os uc u e e ol es by means
o he successi e g ow h o hin ma ensi ic pla es, which
can p opaga e in gi en di ec ions, which co espond o
habi planes, un il hey encoun e an exis ing pla e. An AE
e en (a alanche) is associa ed wi h he o ma ion o each
pla e. The numbe o di ec ions de e mines a ian mul i-
plici y. The model ep oduces scale in a iance o a alanches
du ing he ma ensi ic ansi ions and show ha he c i -
ical exponen cha ac e izing he a alanche size dis ibu-
ion dec eases wi h a ian mul iplici y (g ow h di ec ions)
inc eases.
Finally, we conclude ha a alanche c i icali y in ma ensi ic
ansi ions is la gely de e mined by he symme y change a he
ansi ion and is independen o de ails such as he exis ence
o in e nal s esses ha may in luence he ansi ion pa h.
ACKNOWLEDGMENTS
We acknowledge inancial suppo om he Spanish
Minis y o Science (Ma 2013-40590-P and MAT2015-
69777-REDT). E.K.H.S. is g a e ul o EPSRC (G an No.
EP/K009702/1) and he Le e hulme Founda ion (G an No.
RPG-2012-564) o suppo . RUS acili ies ha e been es ab-
lished and main ained in Camb idge h ough g an s om he
Na u al En i onmen Resea ch Council and he Enginee ing
and Physical Sciences Resea ch Council o G ea B i ain o
MAC (NE/B505738/1, NE/F17081/1, EP/I036079/1).
024102-7
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