Ac a Ma e ialia 255 (2023) 119042
A ailable online 30 May 2023
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Cons an plane shi model: S uc u e analysis o ma ensi ic phases in
Ni
50
Mn
27
Ga
22
Fe
1
beyond non-modula ed building blocks
M. Vinog ado a
a
,
*
, A. Sozino
a
, L. S aka
b
, P. Veˇ
´
a
b
, O. Heczko
b
,
e
, M. Zelený
c
, R. Chulis
d
,
E. L¨
ahde an a
a
, K. Ullakko
a
a
Depa men o Physics, LUT Uni e si y, Lappeen an a, Finland
b
FZU - Ins i u e o Physics o he Czech Academy o Sciences, P ague, Czech Republic
c
Ins i u e o Ma e ials Science and Enginee ing, Facul y o Mechanical Enginee ing, B no Uni e si y o Technology, B no, Czech Republic
d
Ins i u e o Me allu gy and Ma e ials Science, Polish Academy o Sciences, K akow, Poland
e
Facul y o Ma hema ics and Physics, Cha les Uni e si y, P ague, Czech Republic
ABSTRACT
Ma ensi es o Ni-Mn-Ga-based alloys consis o hie a chical winning domains spanning om mic o- o nanoscale. This a ec s he di ac ion pa e n and hus can
dec ease he accu acy o he de e mina ion o he c ys al s uc u e. We p opose a me hod o ob ain di e en ma ensi ic phases in Ni-Mn-Ga-Fe alloy wi h simpli ied
a ian mic os uc u es and domain sizes o mo e han 2 mic ome e s. The use o simpli ied a ian mic os uc u es allows he in luence o nanome e -scale domains
on di ac ion line posi ion o be ci cum en ed and enabls he compa ison o he la ice pa ame e s o non-modula ed (NM), i e-laye ed modula ed (10M), and
se en-laye ed (14M) phases in he same empe a u e ange due o he la ge hys e esis o he in e ma ensi ic ans o ma ions. I is ound ha he sho c ys allo-
g aphic axes in NM, 14M, and 10M ma ensi es a he same empe a u e ha e di e en leng hs. As a esul , equilib ium NM s uc u e building blocks canno be used
o build he c ys al s uc u es o 14M and 10M ma ensi es. Ins ead, we in oduce a cons an plane shi model wi h iden ical shi alues o he nea es planes (110)
along [110] o [110] as a eplacemen o he e agonal building blocks model. The wo k demons a es ha plane shi alues di e d ama ically be ween ma -
ensi es, which ag ees wi h ab ini io calcula ions. The applica ion o he cons an plane shi and ha d sphe e models in modula ed la ices o a omic-le el winning
conside a ions is discussed.
1. In oduc ion
Many ascina ing ea u es ha e been epo ed in Ni-Mn-Ga alloys,
such as a la ge magne ocalo ic e ec , gian magne ic ield-induced
s ain o 6–12%, and ex emely low winning s ess o 0.02–1 MPa
[1–8]. Twinning s ess has been p o en o be empe a u e sensi i e and
dependen on he c ys al s uc u e and ype o win bounda ies [9–13].
The c ys al s uc u es obse ed in Ni-Mn-Ga a e he ypical Heusle
cubic L2
1
s uc u e and low-symme y i e-laye ed modula ed (10M),
se en-laye ed modula ed (14M), and non-modula ed e agonal (NM)
ma ensi es [14]. In he con ex o ma ensi es, i is wo h dis inguishing
ma ensi ic ans o ma ion as ans o ma ion be ween cubic aus eni e
and low symme y ma ensi e phases and in e ma ensi ic ans-
o ma ion be ween a ious ma ensi es. Nume ous s udies ha e been
ocused on ma ensi ic ans o ma ions [15–18], howe e , ewe
esea ch has been de o ed o in e ma ensi ic ans o ma ions [19–23],
and hei na u e hus emains less clea .
The signi icance o he p ecise de e mina ion o c ys al s uc u e o
Ni-Mn-Ga-based magne ic shape memo y alloys becomes e iden
conside ing he uniquely high win bounda y mobili y and unc ionali y
o he ma e ial in magne ic ields. The c ys al s uc u e o modula ed
10M and 14M ma ensi ic phases in Ni-Mn-Ga alloys emains a subjec o
esea ch and discussion, and he in luence o la ice modula ion, which
can be commensu a e o incommensu a e [24–29], on alloy p ope ies is
no ye ully clea .
Khacha u yan e al. [30,31] we e he i s o sugges ha modula ed
ma ensi ic phases can occu in alloys wi h e y low win bounda y
ene gy by nano winning om a mo e s able e agonal phase. This
concep has been applied o a s uc u e p oposal o Ni-Mn-Ga ma -
ensi es, in which he modula ed s uc u es 10M and 14M a e desc ibed
by an al e na ing sequence o nano wins comp ising o NM e agonal
building blocks (TBBs) winned on (101) la ice planes. We e e o such
desc ip ion o s uc u e as he TBB model. The 14M s uc u e, deno ed as
(52)
2
, comp ises i e la ice planes in one o ien a ion and wo planes in
he o he o ien a ion. This is epea ed wice o ul ill he a omic
o de ing. Simila ly, he 10M s uc u e can be conside ed as al e na ing
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (M. Vinog ado a).
Con en s lis s a ailable a ScienceDi ec
Ac a Ma e ialia
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Ac a Ma e ialia 255 (2023) 119042
2
nano wins o wid h h ee and wo la ice planes, deno ed as (32)
2
.
The e o e, each ma ensi ic s uc u e can be buil om basic pe ec ly
e agonal uni s o hei mi o ing pa s o ming nano win bounda ies
[32–35]. Te agonal pa ame e s o hese blocks o all ma ensi ic
s uc u es a e equal o a/2 and c/2 la ice pa ame e s o NM s uc u e.
The simplici y o TBB model enabled i s widesp ead applica ion as an
ini ial app oxima ion in ab ini io calcula ions be o e s uc u al op imi-
za ion [36–39].
The TBB model has also been discussed in expe imen al c ys al
s uc u e in es iga ions, bu i s applica ion o he incommensu a e
modula ed s uc u es is challenging due o limi ed numbe o deg ees o
eedom in basic building uni s [26,29,40–43]. Addi ionally, he adi-
ional s uc u e de e mina ion me hods ha e been challenged by he
in luence o hie a chical win mic os uc u es anging om mic o- o
nanoscale [44–47] in he Ni-Mn-Ga sys em, which o igina es om he
win bounda ies wi h high-mobili y and low-ene gy [48].
To o e come he sho comings o he TBB model, we analyze he
a e age la ice o he ma ensi ic phases and he applica ion o a
nano winnig model o Ni-Mn-Ga-based alloys in his s udy. The basic o
a e age la ice desc ip ion is widely used in c ys allog aphy o modu-
la ed la ices. The de e mina ion o he a e age la ice in ol es he
exclusion o modula ion sa elli es om he di ac ion pa e n, and he
conside a ion o solely he main e lec ions [49–53]. The a e age la ice
concep is widely accep ed o Ni-Mn-Ga-based alloys because i easily
explains he magni ude o de o ma ion du ing s ess-induced o mag-
ne ic ield-induced c ys al eo ien a ion om la ice pa ame e s [3–5,
8–10,23,33]. The a e age la ice has been used in a comp ehensi e
analysis o winning in modula ed Ni-Mn-Ga ma ensi es [11], and he
conclusions o analysis has been empi ically con i med o 10M [11,54]
and 14M [55] ma ensi es. Appendix A explains he de ails o di e en
phases in Ni-Mn-Ga-based sys ems and hei la ice pa ame e s in
di e en coo dina e sys ems commonly used in he li e a u e.
To alle ia e he in luence o hie a chical winning, we examined
ma ensi es in a Ni-Mn-Ga-Fe single c ys al wi h simpli ied a ian mi-
c os uc u es a a ious empe a u es anging om 328 K o 98 K. La ge
he mal hys e esis o in e ma ensi ic ans o ma ions allowed com-
pa ison o he la ice pa ame e s o he non-modula ed and modula ed
10M and 14M phases in he same empe a u e ange. Based on expe i-
men al esul s, we p opose a cons an plane shi (CPS) model as a sig-
ni ican imp o emen o he TBB concep . We also discuss how he ha d
sphe e (HS) model can be used o desc ibe he la ice p ope ies mo e
acu a ely.
2. Me hod
The Ni
50
Mn
27
Ga
22
Fe
1
alloy was chosen o he in es iga ion due o
i s la ge he mal hys e esis in he in e ma ensi ic ans o ma ion [56].
Thus, he la ice pa ame e s o all h ee ma ensi ic s uc u es could be
measu ed a he same empe a u e. The single c ys al ingo s we e g own
by di ec ional solidi ica ion using [001] o ien ed seed c ys al in
Adap aMa L d. (Helsinki, Finland). A hea ea men a 1300 K o ou
days was pe o med o make he ingo s chemically homogenous. Slow
cooling o ambien empe a u e a he a e o 50 K/h was applied o
gua an ee comple e a omic o de ing o he Heusle - ype s uc u e and o
a oid he appea ance o any c acks. Cuboid samples wi h a size o 1.0 ×
3.0 ×20 mm and {100} aces o he pa en cubic phase we e cu a 353 K
om he ingo s by wi e elec ical discha ge machining. Samples we e
g ound and any esidual su ace de o ma ion was emo ed by elec o-
polishing. The ansi ion empe a u es we e de e mined by a low- ield
(
μ
0
H ≈10 mT) AC magne ic suscep ibili y measu emen . Cu ie em-
pe a u e was T
C
=380 K. The specimens exhibi ed sha p ma ensi ic
ans o ma ion a T
M
=(M
s
+M
) / 2 =330 K and e e se ans-
o ma ion a T
A
=(A
s
+A
) / 2 =335 K, whe e M
s
≈M
, A
s
≈A
a e
ma ensi e and aus eni e s a and inish empe a u es, espec i ely. The
ansi ion empe a u es a e compa able o hose epo ed in [56].
In e ma ensi ic ans o ma ions 10M-14M-NM we e iden i ied du ing
cooling and hea ing o he in es iga ed alloy using la ice pa ame e
measu emen and ecip ocal space mapping (RSM).
A PANaly ical Empy ean X- ay di ac ome e equipped wi h Cu
ube, λ =1.5406 Å (K
α
1
), polycapilla y X- ay lens, An on Paa domed
cooling s age DCS 500, Eule ian c adle, and a Pixel3D de ec o was used
o he la ice pa ame e s measu emen and RSM. A hyb id mono-
ch oma o was used o some measu emen s. Fo NM ma ensi e, he
sho es axis was de e mined as a =b and c >a =b. Fo he 10M and
14M ma ensi es c is he sho es axis, c <b < a.
To simpli y a ian mic os uc u es o med a e in e ma ensi ic
ans o ma ions, we cons ained he sho c-axis (cubic coo dina e sys-
em) o he ini ially 10M single- a ian s a e as shown in Fig. 1. The
cons ain allowed he sample o con ac i s sho -axis upon empe a-
u e dec ease bu p ohibi ed new a ian s wi h long c ys allog aphic
axes along he cons ained di ec ion. The sample was placed in a 0.05
mm hick coppe oil ha was olded o e a bo h ends o he sample. In
his con igu a ion, he sho c ys allog aphic axis was pa allel o he
longes side o he sample and pe pendicula o he di ac ion plane.
The sho c ys allog aphic axis was ixed and could no eo ien du ing
in e ma ensi ic ans o ma ions, which means ha new a ian s wi h a
long axis along he longes side o he sample could no appea . Thus, we
in es iga ed long axes in all ma ensi es using he same di ac ion plane
wi hou any need o sample il ing o o a ion.
Op ical obse a ions o win- a ian mic os uc u es we e aken
wi h he cons ained sample ins alled in a he mal chambe wi h a
anspa en window (Linkam DSC 600). The sample was cooled om
413 K o 120 K a a a e close o 4 K/min. The di e en ial in e e ence
(Noma ski) con as in a Zeiss AxioImage .Z1 mic oscope showed
di e en su ace il s o indi idual win domains as egions wi h
di e en colo s.
3. Resul s
The p esump ion o simpli ied a ian mic os uc u e in he con-
s ained s a e was i s alida ed by op ical obse a ions. The mic o-
g aphs o he sample su ace a selec ed empe a u es a e shown in
Fig. 2. Ini ially, he 10M sample a oom empe a u e consis ed o a
nea ly single a ian , which con ained one na ow esidual (modula-
ion) win, Fig. 2a. This 10M nea ly single a ian wi h espec o he c-
axis (modula ion and a/b wins s ill exis ed) ans o med in o a wo-
a ian 14M ma ensi e. In mic og aphs, he a ian s di e by hei
colo s. All win domain aces we e pa allel o he cons ained sho axis
wi h he domain size anging om 2 o 100
μ
m. Such a ian mic o-
s uc u e simpli ied conside ably he in e p e a ion o di ac ion e-
sul s. Unde u he cooling, he wo- a ian 14M ma ensi e s a ed o
ans o m in o a wo- a ian NM s uc u e, Fig. 2e. Due o he empe -
a u e limi , we we e no able o comple e he ans o ma ion o he NM
phase unde op ical obse a ion. Howe e , in he pa ial 14M-NM
ans o ma ion, we obse ed simila win domain scale and he same
ace o ien a ion along he cons ained axis. This gi es easonable
Fig. 1. Schema ic illus a ion o 10M sample wi h cons ained c-axis, c <a, b.
Axes a and b a e no dis inguished due o a/b winning.
M. Vinog ado a e al.
Ac a Ma e ialia 255 (2023) 119042
3
con idence ha he inal NM s uc u e would consis o only wo
a ian s.
In o de o p o ide mo e eliable e idence, he RSM was used o
e i y he c ys al s uc u e and o ien a ion o he sample (Fig. 2b, d, ).
Two- a ian mic os uc u es we e con i med o 14M and NM ma -
ensi es. A nea ly single modula ion di ec ion was ound in he ini ial
Fig. 2. Op ical images o he sample su ace (100) showing simpli ied ma ensi e mic os uc u es. The o ien a ion o he sho axis in he image plane is indica ed by
he a ows. (a) 10M (303 K), (c) 14M (223 K), (e) a mix u e o 14M and NM ma ensi es (120 K). Recip ocal space maps o a plane pe pendicula o he sho axis o
(b) 10M (303 K), (d) 14M (223 K), and ( ) NM (98 K) ma ensi es a e in e ma ensi ic ans o ma ions du ing cooling.
Fig. 3. (a) Tempe a u e dependence o la ice pa ame e s in cubic coo dina es measu ed a he cons ained condi ion o 10M, 14M, and NM ma ensi es wi h he
sho axis pe pendicula o he di ac ion plane. (b) Tempe a u e dependence o he sho c ys allog aphic axis leng h o 10M, 14M, and NM ma ensi es.
M. Vinog ado a e al.
Ac a Ma e ialia 255 (2023) 119042
4
10M ma ensi e s uc u e and his modula ion di ec ion was inhe i ed
by he 14M phase.
Fo 10M ma ensi e (Fig. 2b), (400) peak (and (040) peak due o a/b
wins) is isible in he cen e o he ma ked a ea ( ed ec angle), which
co esponds o he sample o ien a ion wi h he a,b-axis e ical in he
di ac ion plane. Fou sa elli es (S
1
400
, S
2
400
, S
-2
620
, S
-1
620
) a e clea ly isible
be ween he (400) and (620) peaks, con i ming he 10M c ys al
s uc u e. Lowe -in ensi y sa elli es a e isible in a pe pendicula di-
ec ion, indica ing he p esence o modula ion wins in a small amoun .
The ans o ma ion om 10M o 14M ma ensi e (Fig. 2d) esul s in he
disappea ance o he 10M (400) peak and he appea ance o wo new
peaks. These wo peaks co espond o wo 14M a ian s 1 and 2,
iden i ied as win- ela ed peak pai s (400)
1
and (040)
2
, and (620)
1
and (260)
2
, based on 2θ angles o bo h peaks and he angle be ween he
Fig. 4. Tempe a u e dependence o (a) 2θ angles o (444) and (
444) peaks, and (b) d-spacings o indi idual ma ensi ic phases.
Fig. 5. Schema ic ep esen a ion o he CPS model o a ma ensi e wi h modula ed la ice. (a) Compa ison o CPS and TBB models; colo dis inguishes ype o a oms
in h ee componen alloy. (b) View o he wo nea es (110) planes wi h a plane no mal ou o pape ; colo dis inguishes he a angemen o a oms in planes no he
ype o a om. The shi be ween he nea es (110) planes is dx. 3D iew o he nea es (110) planes: (c) in a cubic la ice, (d) a e he e agonal dis o ion along he c-
axis, and (e) a e he e agonal dis o ion along c-axis o 10M o 14M ma ensi e la ice (c <a), and plane shi .
M. Vinog ado a e al.
Ac a Ma e ialia 255 (2023) 119042
5
planes. The di ec ion o each plane no mal is in he di ac ion plane and
is de e mined by he alue o
ω
(angle be ween he inciden beam and
ho izon al in he di ac ion plane) and 2θ angles. The es o he peaks
a e sa elli es o bo h a ian s. The dis ance be ween he main peaks o
a ian 1 and a ian 2 is almos equal o he dis ance be ween he
nea es sa elli es. Two- a ian 14M ma ensi e ans o ms o wo-
a ian NM ma ensi e (Fig. 2 ). These wo a ian s we e iden i ied as
peak’s pai s (004)
1
and (400)
2
, and (206)
1
and (602)
2
, based on 2θ
angles o bo h peaks and he angle be ween he planes. Sa elli es con-
nec ed o he la ice modula ion in 10M and 14M ma ensi es disappea
in he NM phase.
Figu e 3a shows he empe a u e dependence o he la ice pa am-
e e s in cubic coo dina es o all ma ensi ic phases, measu ed unde
cons ained c-axis condi ion. The p esen ed da a is a combina ion o
cooling and hea ing measu emen s. The empe a u e ange o he in-
s umen limi s he da a o NM ma ensi e a low empe a u es, while
NM-14M in e ma ensi ic ans o ma ion limi s da a a high empe a-
u es. The da a o 14M ma ensi e is limi ed by 14M-NM a low em-
pe a u es and 14M-10M in e ma ensi ic ans o ma ions a high
empe a u es. The da a o 10M is limi ed by he 10M-14M in e -
ma ensi ic ansi ion a low empe a u es and he 10M-cubic ma ens-
i ic ans o ma ion a high empe a u es. The he mal hys e esis o
in e ma ensi ic ans o ma ions is conside ably highe in compa ison o
ma ensi ic hys e esis, which is abou 5 K be ween aus eni e and
ma ensi e. Fo 10M-14M ans o ma ion, he hys e esis is abou 60 K,
and o 14M-NM ans o ma ion 90 K.
Di ec de e mina ion o he a- and b-la ice pa ame e s om (400)
and (040) in 10M ma ensi e is p oblema ic due o hei small di e ence.
To o e come his, we measu ed 2θ o (602), (062) and (004) peaks o
calcula e he a- and b-axis la ice pa ame e s. The 2θ angle posi ions o
(602) and (062) peaks a e high and do no depend on la ice monoclinic
dis o ion, and he calcula ion is hus su icien ly p ecise and s aigh -
o wa d. The di e ence be ween a- and b- pa ame e s inc eased wi h
dec easing empe a u e, in co espondence wi h p e ious epo s (e.g.,
[17]), bu he peaks me ged a empe a u es below 263 K. Incommen-
su a e modula ion [56] o a/b winning [44], a e conside ed as possible
mechanisms o such a di ac ion e ec .
The 2θ angle o peaks (004) o modula ed ma ensi es and (400) o
NM we e measu ed o de e mine he sho axis la ice pa ame e . Fo he
NM ma ensi e sho -axis measu emen s we e done in a ian 2 o he
wo- a ian s a e du ing hea ing om 98 K. Fo he sho -axis mea-
su emen s in 10M ma ensi e, he sample was eo ien ed, hus he sho
c ys allog aphic c-axis was in he di ac ion plane. Measu emen s we e
done du ing cooling om 328 K. A e 10M-14M in e ma ensi ic
ans o ma ion a 193 K one a ian o he 14M wo- a ian s a e
e ained di ec ion o he c-axis in he di ac ion plane. This way, he
sho -axis o he 14M ma ensi e was measu ed in he empe a u e ange
o he 14M phase s abili y.
Figu e 3b shows ha he sho axis changes ab up ly du ing he
in e ma ensi ic ans o ma ions. Fo 10M and 14M he c-axis is he
sho es , o NM he sho es a e wo axes a =b. The leng h o he sho
axis is di e en o all h ee phases a he same empe a u e and in-
c eases wi h empe a u e ise.
The (444) and (444) peak posi ions de e mine d-spacings d
111
and
d111 wi h high accu acy due o he high 2θ angles. Figu e 4a shows he
empe a u e dependence o he 2θ angles o (444) and (444) peaks. The
2θ di e ence o he (444) and (444) peaks posi ions is highly sensi i e o
la ice monoclinic dis o ion. The di e ence and co esponding mono-
clinic dis o ion a e highes in he 14M phase and decline wi h
inc easing empe a u e o bo h 14M and 10M phases. The e is no
monoclinic dis o ion in he e agonal NM phase, hus he (444) and
(444) peaks o e lap. The expe imen al da a o he c-axis and d-spacings
d
111
and d111 we e used o de e mine d110 and d110, based on geome ical
conside a ions o 10M and 14M c ys al la ices:
d110 =c

(c
d111)2
−1
√(1)
d110 =c

(c
d111)2
√−1
(2)
The co esponding empe a u e dependences o hese d-spacings a e
gi en in Fig. 4b. In 10M and 14M modula ed ma ensi es, d
110
/2 co -
esponds o he dis ance be ween he closes planes. The (110) planes
a e shi ed ela i e o each o he , causing la ice modula ion. Using he
cubic coo dina e sys em (Appendix, Table A1) he c-axis is he sho es
in 10M and 14M, bu he longes in NM. This p o ides con usion when
compa ing he s uc u es. The e o e, o a oid i , he (101) spacing in
NM, which is di ec ly ela ed o he concep o cons an plane shi , was
calcula ed by eplacing d
110
wi h d
101
in Eq. (1). Such an ope a ion al-
lows us o compa e he d-spacings o he shi ing planes o all
ma ensi e s uc u es. In NM and 14M ma ensi e, he empe a u e de-
pendency o d
110
(14M) and d
101
(NM) is negligible, whe eas i is weak
o 10M. Simila negligible dependence o d
110
on he s uc u e ype was
obse ed in TEM s udy [57].
3.1. Cons an plane shi (CPS) model
I is appa en om ou measu emen s ha he applica ion o a TBB
concep o 14M and 10M ma ensi es does no wo k wi h espec o he
la ice pa ame e s. The main a gumen is ha he sho c ys allog aphic
axis is di e en o each ma ensi e a he same empe a u e (Fig. 3b).
This means ha he equilib ium e agonal c/a a io o NM ma ensi e
canno be used o build modula ed phases by simple geome ical
Fig. 6. Tempe a u e dependence o he plane shi dx o indi idual ma ens-
i ic phases.
Table 1
Space diagonals SD, SD
sho
along [111] and SD
long
along [111] in di e en
phases o Ni-Mn-Ga alloy. They a e equal in cubic and NM e agonal la ice, and
sligh ly di e en in monoclinic 10M and 14M la ices.
T =273 K T =193 K
SD, Å SD, Å
Cubic 10.099* 10.088*
NM 10.161* 10.157
SD
long
, Å SD
sho
, Å SD
long
, Å SD
sho
, Å
10M 10.131 10.076 10.133* 10.061*
14M 10.137 10.065 10.137 10.050
*Ex apola ed alues.
M. Vinog ado a e al.

Ac a Ma e ialia 255 (2023) 119042
6
ope a ions as sugges ed in [32–35]. I is possible o conside a ious
kinds o a la ice elaxa ion di e en in each ma ensi ic s uc u e [58],
bu hen he simplici y o he TBB concep would be los . Thus, he e is
s ill missing a uni ied model, which is able o desc ibe a ious modu-
la ions o ma ensi ic s uc u es by a single pa ame e . The e o e, we
in oduce a cons an plane shi (CPS) model desc ibed in Fig. 5, as an
ex ension o he TBB concep . The model p oposes a cons an shi dx o
he plane (110) along [110] o [110] di ec ions in acco dance wi h he
well-known (32)
2
sequence o 10M ma ensi e and (52)
2
sequence o
14M ma ensi e, as alida ed by ansmission elec on mic oscope
(TEM) s uc u al s udies [57,59]. The CPS model, unlike he TBB
concep , does no assume no equi e he same sho axis leng h in he
NM, 14M, and 10M ma ensi es.
Figu e 5a shows he di e ence be ween he TBB and CPS models. In
he TBB model, he modula ed la ice o 10M o 14M ma ensi es is
made om he building block o NM la ice, which is shown in Fig. 5a as
a ec angle wi h sizes a
NM
/2, c
NM
/2. Mi o ing building block a he
nano win bounda ies c ea es he modula ed la ice. The CPS model
akes one ea u e om TBB model: he cons an absolu e alue o he
plane (110) shi , |dx|, u he e e ed o as cons an dx shi o jus dx.
O he TBB model p ope ies, like b
NM
=c
10M
and dx
NM
=dx
10M,
do no
ma ch up wi h la ice pa ame e measu emen s, as shown in Fig. 3b and
addi ionally discussed below. Fu he mo e, ou calcula ions demon-
s a e ha he angle o building blocks β’ di e s om he alue o 90◦by
abou 0.15◦ o 0.6◦. This a ia ion depends on he empe a u e and he
ype o modula ed la ice. The ad an age o he CPS model is ha he dx
shi can be ound expe imen ally, using he measu ed la ice pa ame-
e s a, b, and c, and d-spacing d
110
and d110. In cu en in es iga ions, we
calcula ed he d-spacings om measu ed 2θ angles o (444), (444), and
(004) using Eqs. (1) and (2).
The me hod o de e mining he dx shi o all ma ensi es is shown
below. The spacings d
110
and d110 a e calcula ed om he measu ed c-
axis and d
111
and d111 based on Eqs. (1) and (2). The (52)
2
sequence o
14M ma ensi e and (32)
2
sequence o 10M ma ensi e de e mine he
angle be ween diagonals in he pa allelog am build using he la ice
pa ame e s a and b (o a and c o NM) (see Appendix A, Fig. A2) as:
γ10M
diag =
π
2− an−1(2dx
5d110)(3)
γ14M
diag =
π
2− an−1(6dx
7d110)(4)
NM ma ensi e also can be desc ibed using CPS as:
γNM
diag =
π
2− an−1(2dx
d101)(5)
The leng hs o he hal diagonals (D1, D2) in he pa allelog am build
using he la ice pa ame e s a and b a e de e mined by he equa ions:
D1=d110
sin(γdiag)(6)
D2=d110
sin(γdiag),(7)
whe e γ
diag
is subs i u ed by γ10M
diag ,γ14M
diag o γNM
diag and d
110
is subs i u ed o
d
101
o NM.
Equa ions (3)-(7) de e mine he alues o he hal diagonals D1 and
D2, as well as he angle γ
diag
be ween hem. They depend on he shi dx
and he espec i e sequence, (32)
2
o 10M and (52)
2
o 14M
ma ensi e. The known D1, D2, and γ
diag
allow us o calcula e he la ice
pa ame e s a and b (a and c o NM) using known igonome ic e-
la ions. By a e e se p ocedu e, i.e., by minimizing he di e ence be-
ween he calcula ed and measu ed alues o la ice pa ame e s, we
ound dx o each empe a u e and ype o ma ensi e. Figu e 6 shows
he empe a u e dependence o he plane shi dx. I can be seen ha dx
inc eases wi h dec easing empe a u e o all ma ensi es. The plane
shi is signi ican ly di e en in di e en ma ensi es. I is lowes in 10M
ma ensi e, inc eases h ee old upon 10M-14M ans o ma ion, and
inally inc eases by u he 30% du ing 14M-NM ans o ma ion.
We compa ed he expe imen ally de e mined dx wi h he co e-
sponding shi s calcula ed om la ice cons an s ob ained by ab ini io
calcula ions wi h he elec on delocaliza ion co ec ion [60]. The TBB
model was used o c ea e ini ial s uc u es, which we e u he ully
op imized o ob ain he lowes ene gy. Howe e , he de ails abou
cha ac e o modula ion o ully op imized s uc u es we e no discussed
in Re . [60]. The ab ini io alues o dx calcula ed wi h help o Eqs. (3)–(5)
a e 0.042 o 10M, 0.282 o 14M and 0.346 o NM s uc u e. The
ag eemen is e y good o all h ee ma ensi ic s uc u es al hough he
calcula ions we e pe o med o s oichiome ic Ni
2
MnGa composi ion
and equilib ium la ice pa ame e s co esponded o condi ions a 0 K.
Using he ab ini io la ice cons an s calcula ed wi hou delocaliza ion
co ec ion, he alues o dx a e nea ly he same, in ange 0.486–0.52, o
all h ee s uc u es [60]. This compa ison suppo s he necessi y o he
elec on delocaliza ion co ec ion o ab ini io calcula ions o Ni-Mn-Ga
alloys. Mo eo e , i also sugges s ha epo ed di e ences in dx a e
basic p ope ies o ma ensi ic la ices and a e independen on empe -
a u e o alloy composi ion. Co ec ed ab ini io calcula ions also p o ide
nea ly cons an alues o in e laye dis ances d
110
(10M, 14M) and d
101
(NM) calcula ed wi h help o Eq. (7). They change o all h ee
ma ensi ic s uc u es only om 4.22 Å o 4.23 Å, in a good ag eemen
wi h simila nea ly cons an dependency in expe imen , Fig. 4b.
We ound ha he CPS model and ou obse a ions can be linked
di ec ly wi h well-known ha d sphe e (HS) model o BCC- ype la ice.
The HS model was p e iously used, e.g., o desc ip ion o winning in
magnesium [61]. In HS model, he highes Young’s modulus is in he
di ec ions o space diagonals ma ked as ed lines in Fig. 5c. The nea es
a oms ouch one ano he along cube diagonals [111], and uni cell
leng h a and a omic adius R a e ela ed h ough 4R/
3
√in cubic la ice.
Ou calcula ions, based on explici exp essions p esen ed in Re . [62]
Eqs. (4) and ((5) in he e e ence), and measu ed elas ic coe icien s
[63–67], con i m he highes Young’s modulus E
111
in cubic as well as in
ma ensi ic phases o he Ni-Mn-Ga alloys.
Addi ional link be ween CPS and HS models comes om ou la ice
pa ame e s measu emen s. We calcula ed la ice space diagonal SD
along he [111] di ec ion o cubic and NM phases, as well as o he
monoclinic la ices 10M and 14M wi h wo di e en diagonals, sho
SD
sho
along he [111] and mo e long SD
long
along he [111]. Da a o
Table 1 we e de e mined a wo empe a u es, 273 K and 193 K. Fo he
14M ma ensi e, he la ice pa ame e s we e measu ed di ec ly a bo h
empe a u es. Fo he 10M ma ensi e, pa ame e s a 273 K we e
measu ed and pa ame e s a 193 K we e calcula ed using linea
ex apola ion. Fo he NM ma ensi e, pa ame e s a 193 K we e
measu ed and pa ame e s a 273 K we e calcula ed using linea
ex apola ion. The cubic phase exis s abo e 333 K; hus, o compa e his
phase wi h o he s, la ice pa ame e s we e measu ed wi hin he ange
333 – 723 K, which was ollowed by linea ex apola ion o 193 K. I is
seen om he Table 1 ha he di e ence in SD alues be ween phases is
e y small, less han 0.5%, which indica es ha he sho diagonal is
di icul o comp ess u he , and s ongly suppo s he alidi y o he HS
model.
Fo he HS model, a small e agonal dis o ion o he cubic la ice
wi h he cons an sho es dis ance be ween a oms esul s in equal
expansion o he la ice in wo pe pendicula di ec ions. This is shown in
Fig. 5c and d. Shi ing o he (110) planes ela i e o each o he esul s in
sho e dis ance be ween he planes, d
110
/2 <D1/2, because planes wi h
cons an sho es in e a omic dis ances a e inclined om 90◦, Fig. 5e.
Thus, he HS model conside a ions gi e a plausible physical explana ion
why he expe imen al esul s show ha d
110
<d110 o la ices wi h
modula ions (Fig. 4b). E en hough he (110) plane is shi ed, he D1
M. Vinog ado a e al.
Ac a Ma e ialia 255 (2023) 119042
7
(see Fig. 5e) does no change since he ha d di ec ions in he plane keep
he plane shape. Fu he mo e, he coupling o he CPS and HS models
allows o he p edic ion o Poisson coe icien s. The s ess-induced
de o ma ion o a modula ed la ice along he c-axis e lec s as app oxi-
ma ely he same de o ma ion pe pendicula o c-axis, indica ed as D1 in
Fig. 5e. Thus, o his pai o di ec ions, he Poisson a io will be close o
one. In con as , he e will be a minimum de o ma ion along he plane
(110) no mal, due o (110) plane shi ing, and he co esponding Pois-
son a io will be close o ze o. In addi ion o his, he pe iodic shi ing o
he (110) planes c ea es a pe iodic change in posi ion o a oms in he a-c
o b-c winning plane (101) o (011) in he di ec ion o he [110]. The
win planes, ei he (101) o (011), change om a smoo h la su ace o
being a saw oo h-shaped modula ed su ace. This modula ion wa e
should in e ac wi h he win bounda y mo ion by changing locally dx
alue o he (110) planes. I means ha a highe numbe o a oms is
in ol ed in a/c o b/c win bounda y mo ion in modula ed s uc u es in
compa ison wi h la ices wi hou modula ion. Mo e esea ch on his
issue is needed o unde s and why Ni-Mn-Ga-based sys ems ha e such
low winning s ess, making hem one o he mos in e es ing magne ic
shape memo y ma e ials.
4. Conclusion
•An expe imen al me hod is p oposed ha simpli ies a ian mic o-
s uc u e a e in e ma ensi ic ans o ma ions by cons aining he
sho es c ys allog aphic axis. In Ni-Mn-Ga-Fe alloy, he ini ial nea
single- a ian 10M s uc u e wi h cons ained c-axis ans o med
in o 14M and hen u he o NM wo- a ian mic os uc u es wi h
domain sizes anging om 2 o 100
μ
m.
•The simpli ied a ian mic os uc u e enabled con iden de e mina-
ion o la ice pa ame e s o 10M, 14M, and NM ma ensi es wi hin
he empe a u e ange be ween 328 K and 98 K. I is shown ha all
p incipal axes change discon inuously du ing he in e ma ensi ic
ans o ma ions.
•We in oduced a cons an plane shi (CPS) model o e lec p ecisely
he con inuous and discon inuous changes in he la ice pa ame e s
wi h empe a u e. The model p oposes equal shi alue o he
nea es planes (110) along [110] o [110] di ec ions.
•The plane shi is signi ican ly di e en in di e en ma ensi es. I is
lowes in 10M ma ensi e, inc eases abou h ee imes upon 10M-
14M ans o ma ion, and inally inc eases by a u he 30% du ing
14M-NM ans o ma ion.
•We p opose ha he combina ion o CPS and ha d sphe e model can
be use ul o a omic-le el conside a ions o winning and unde -
s anding o elas ic p ope ies in modula ed la ices.
Decla a ion o Compe ing In e es
The au ho s decla e ha hey ha e no known compe ing inancial
in e es s o pe sonal ela ionships ha could ha e appea ed o in luence
he wo k epo ed in his pape .
Acknowledgmen s
This wo k was inancially suppo ed by he Academy o Finland
(g an numbe 325910). The au ho s om he Czech Republic
acknowledge he unding om he Czech Science Founda ion g an no.
21-06613S and om he ESIF and MEYS p ojec s Ma Fun – CZ.02.1.01/
0.0/0.0/15 003/0000487 and SOLID21 – CZ.02.1.01/0.0/0.0/16_019/
0000760. The in e na ional mobili y o Alexei Sozino was suppo ed by
ESIF and MEYS p ojec MOBILITY FZU 2 – CZ.02.2.69/0.0/0.0/18_053/
0016627. Robe Chulis acknowledges he p ojec s 2021/42/E/ST5/
00367 o he Na ional Science cen e o Poland. We hank Sil ia
Sedl´
ako ´
a o he kind assis ance wi h he ope a ion o Linkam DSC600
he mal chambe .
Appendix A. A e age la ices in Ni–Mn–Ga-based sys ems
This Appendix explains he di e en phases in Ni-Mn-Ga-based sys-
ems and hei la ice pa ame e s in di e en coo dina e sys ems
commonly used in he li e a u e. The cubic phase obse ed in
Ni
50
Mn
27
Ga
22
Fe
1
is he ypical Heusle L2
1
s uc u e. The cubic phase is
desc ibed in Fig. A1. The la ice pa ame e s o he low-symme y i e-
laye ed modula ed (10M), se en-laye ed modula ed (14M), and non-
modula ed e agonal (NM) ma ensi es [14] a 273 K a e desc ibed in
de ail in Table A1. Fo he 10M and 14M phases, he la ice pa ame e s
we e measu ed di ec ly. The cubic phase and NM phase pa ame e s we e
calcula ed using linea ex apola ion. A e sample cooling om he
cubic phase, he modula ion in 10M was commensu a e, bu i became
incommensu a e below 263 K and emained in his s a e a e hea ing
om low empe a u es o empe a u es close o he e e se ma ensi e
ans o ma ion (see de ails in [56]).
We used he a e age la ice (o iden ically a e age s uc u e) o he
ma ensi ic phases o analyze he applica ion o a nano winnig model o
Fig. A1. Cubic Heusle s uc u e Ni
2
MnGa. (a) – Schema ic ep esen a ion o he L2
1
uni cell. (b) – The ela ion be ween he cubic and diagonal coo dina es; (c) –
The p ojec ion o a omic s uc u e o he plane.
M. Vinog ado a e al.
Ac a Ma e ialia 255 (2023) 119042
8
Ni-Mn-Ga-based alloys and o de elop he CPS model in his s udy. The
a e age la ice (a e age s uc u e) is widely used in c ys allog aphy o
he desc ip ion o bo h commensu a e and incommensu a e (C and IC)
modula ed s uc u es. The a e age la ice is ypically de e mined om
he di ac ion pa e n conside ing only he main e lec ions and
igno ing he sa elli es ela ed o modula ion. In oducing he supe space
[49,50] es o es he ansla ional symme y in he c ys als lacking he
3D pe iodici y (IC s uc u e). The same app oach can be used o he C
s uc u e. The a e age la ice app oach is ex emely bene icial o he
case o he s udied ma e ial, in which he modula ion changes wi h
empe a u e be ween C and IC. I signi ican ly simpli ies he conside -
a ions o mos physical mechanisms ela ed o he la ice geome y, such
as la ice eo ien a ion o winning.
We no e ha in he case o commensu a e s uc u e, c ys al la ice
can al e na i ely be desc ibed by a new la ge uni cell, i.e., he c ys al
consis s o iden ical sec ions o modula ed o iginal cells. These uni cells
a e commonly e e ed o as long pe iodical (LP) cells o supe cells. In
his case, all he e lec ions o he di ac ion pa e n a e indexed using
h ee indices, whe eas mo e indices a e equi ed o IC s uc u es
[49–52].
The a e age la ice o modula ed ma ensi es 10M and 14M in cubic
coo dina es, coinciding wi h [100], [010], and [001] o L2
1
Heusle
supe s uc u e, is monoclinic and widely used o in es iga ions o Ni-
Mn-Ga-based ma ensi es. The la ges s ess- o magne ic ield-induced
s ain co esponds o he di e ence in la ice pa ame e s o he
a e age la ices [3–5,8–10,23,33]. A e phase ans o ma ion, he
o ien a ion o he a e age la ice axes di e s only by a ew deg ees om
he cubic phase di ec ions [100], [010], and [001] in win a ian s o
10M and 14M modula ed ma ensi es. As a esul , he maximum s ess
du ing c ys al eo ien a ion is eco ded along cubic coo dina es. A e a
de ailed in es iga ion o winning in modula ed Ni-Mn-Ga [11],
employing a e age la ice in cubic coo dina es and empi ical p oo o
10M [11,54] and 14M [55] ma ensi es, he a e age la ice app oach
u ns e en mo e use ul. A e age la ice in cubic coo dina es can be
Table A1
La ice pa ame e s o ma ensi ic phases o Ni
50
Mn
27
Ga
22
Fe
1
alloy a 273 K in di e en coo dina e sys ems.
10M 14M NM
A e age la ice in cubic (black) and diagonal coo dina es ( ed).
Cubic:
a ≅b >c
a =5.989 Å
b =5.942 Å
c =5.560 Å
(1 - c/a) =0.072
γ
cubic
=90.45 deg
Diagonal:
D1 =4.235 Å
D2 =4.202 Å
γ
diag
=89.55 deg
Poin g oup: 2/m
Cubic:
a >b >c
a =6.146 Å
b =5.803 Å
c =5.529 Å
(1 - c/a) =0.10
γ
cubic
=90.58 deg
Diagonal:
D1 =4.248 Å
D2 =4.205 Å
γ
diag
=86.71 deg
Poin g oup: 2/m
Cubic:
a =b <c
a =b =5.503 Å
c =6.529 Å
(c/a −1) =0.19
γ
cubic
=90 deg
Space g oup: I4/mmm
(No.139)
Diagonal (2M [35,69]):
D1 =D2 =4.269 Å
γ
diag
=80.26 deg
Commensu a e s uc u es in diagonal coo dina es.
A oms in 2D s uc u es a e p ojec ion o wo nea es planes.
Dashed lines indica e nano win bounda ies in TBB model.
Di ec ion indices show o ien a ion o cubic coo dina e sys em.
Long-pe iodical (3
2)
2
|dx| =0.0828 Å
a
LP
=D1 =4.235 Å
b
LP
=5*D2 =21.009 Å
c =5.560 Å
γ
diag
=89.55 deg
Long-pe iodical (5
2)
2
|dx| =0.2829 Å
a
LP
=D1 =4.248 Å
b
LP
=7*D2 =29.436 Å
c =5.529 Å
γ
diag
=86.71 deg
2M monoclinic[35,69]
|dx| =0.3609 Å
D1 =4.269 Å
D2=4.269 Å
b =5.503 Å
γ
diag
=80.26 deg
Commensu a e s uc u e in con en ional c ys allog aphic coo dina es
(b-axis pe pendicula o bo h a- and c-axis).
Long-pe iodical
a =4.235 Å
b =5.560 Å
c =21.009 Å
β =89.55 deg Space g oup: C2/m
(No.12)
Long-pe iodical
a =4.248 Å
b =5.529 Å
c =29.436 Å
β =86.71 deg Space g oup: C2/m
(No.12)
2M monoclinic
a =c =4.269 Å
b =5.503 Å
β =80.26 deg
M. Vinog ado a e al.
Ac a Ma e ialia 255 (2023) 119042
9
ans o med o a diagonal coo dina e sys em, wi h he la ice axes coin-
ciding wi h diagonals in a pa allelog am buil on cubic a- and b-axes o
modula ed la ices (see Fig. A2).
The diagonal coo dina e sys em is bene icial o examining long
pe iodic la ices wi h commensu a e modula ions ha a e i e and se en
imes longe in he di ec ion o modula ion o 10M and 14M, espec-
i ely. NM ma ensi e in cubic coo dina es is simply e agonally dis-
o ed L2
1
along [001] di ec ion. We made a speci ic decision on he
diagonal coo dina es o he NM ma ensi e. We conside diagonals D1
and D2 in he plane including a-axis and c-axis o seamless compa ison
wi h laye ed s uc u es. Such p esen a ion o NM la ice is well known in
he li e a u e as 2M "monoclinic’’ [35,68,69]. This me hod enabled us o
p o ide plane shi dx o he NM la ice in a simila way as o laye ed
s uc u es - by minimizing he di e ence be ween he calcula ed and
measu ed alues o la ice pa ame e s a and b (a =b o NM ma ensi e
in cubic coo dina es).
The ela ion o a e age la ice axes wi h con en ional c ys allo-
g aphic coo dina es sys em, whe e he b-axis is pe pendicula o he a-
and c-axis and monoclinic angle is β, p esen ed in Tables A1 and A2. The
ables also p o ide de ails ega ding he space g oups p oposed in
a ious s udies, including o 10M and 14M monoclinic ma ensi e la -
ices. Bo h s uc u es may be commensu a e o incommensu a e [24,
25]. Fo commensu a e s uc u es a omic coo dina es a e he same o
all empe a u es, whe eas o incommensu a e s uc u es modula ion
ec o is empe a u e dependen [28,56], hus he a omic posi ions a e
also a ec ed by empe a u e.
Table A2 shows he a e age incommensu a e la ice pa ame e s o
10M ma ensi e in di e en coo dina es. A low empe a u es he di -
e ence be ween a and b axes disappea s (see Figs. 3a and 6). When a =b
in cubic coo dina es, D1 is pe pendicula o D2 in diagonal coo dina es,
bu no equal. Simul aneously, he s uc u e becomes incommensu a e,
as e idenced by he sa elli e posi ions de ec ed by Veˇ
´
a e al. [56] and
e i ied by ou p esen esea ch (da a no p esen ed in he ex ).
Cu en in es iga ion sugges s a simple algo i hm o calcula e a omic
coo dina es in modula ed commensu a e s uc u es 10M and 14M,
howe e , he case o incommensu a e s uc u es is ou o he scope o
his wo k.
Re e ences
[1] P.J. Webs e , K.R.A. Ziebeck, S.L. Town, M.S. Peak, Magne ic o de and phase
ans o ma ion in Ni 2 MnGa, Philos. Mag. B. 49 (1984) 295–310, h ps://doi.o g/
10.1080/13642817408246515.
[2] K. Ullakko, Magne ically con olled shape memo y alloys: a new class o ac ua o
ma e ials, J. Ma e . Eng. Pe o m. 5 (1996) 405–409, h ps://doi.o g/10.1007/
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Table A2
A e age la ice o 10M incommensu a e s uc u e.
Coo dina e sys em Cubic Diagonal Con en ional c ys allog aphic
La ice pa ame e s a =b =5.967 Å
c =5.560 Å
γ
cubic
=90.43 deg
D1 =4.235 Å
D2 =4.203 Å
c =5.560 Å
γ
diag
=90 deg
a =4.235 Å
b =5.560 Å
c =4.203 Å
β =90 deg
Supe space g oup Immm(00γ)s00 [24]
Fig. A2. Rela ions be ween la ice pa ame e s in cubic and diagonal coo dina es o 10M, 14M and NM ma ensi es. D1 and D2 a e hal diagonals in he pa al-
lelog am build using he la ice pa ame e s. (a) 10M and 14M modula ed ma ensi es; (b) NM ma ensi e. The indices o d-spacings a e in cubic coo dina es.
M. Vinog ado a e al.