RESEARCH ARTICLE
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Influence o Hys e esis on Magne ocalo ic Pe o mance a
C yogenic Tempe a u es: A Tb3Ni Case S udy
Timo Niehoff,* Benedik Beckmann, Kons an in Skoko , A i z He e o, Albe o Oleaga,
Edua d Byko , Ca alina Salaza Mejía, Ma c S aßheim, Oli e Gu fleisch, J. Wosni za,
and Tino Go schall
The magne ocalo ic effec (MCE) offe s a p omising al e na i e o
en i onmen ally iendly cooling echnologies, pa icula ly a c yogenic
empe a u es. Howe e , o e es ima ing ma e ial capabili ies can lead o
misguided esea ch effo s and hinde echnological p og ess. Me amagne ic
ma e ials unde going a ansi ion om an an i e omagne ic o a
e omagne ic s a e a e o en p edic ed o exhibi a s ong in e se MCE a
c yogenic empe a u es based on magne iza ion measu emen s. This
assump ion is c i ically assessed he e using Tb3Ni as a case s udy. By
employing a simple model and compa ing esul s ac oss a ious
measu emen echniques, i is demons a ed ha he p edic ed in e se MCE
does no exis . Specific-hea da a e eal no e idence o his effec , while di ec
𝚫Tad pulsed-magne ic-field measu emen s indica e significan hea ing caused
by dissipa i e effec s linked o hys e esis. Fu he mo e, o al-en opy
calcula ions de i ed om magne iza ion da a iola e he second law o
he modynamics, clea ly uling ou he exis ence o an in e se MCE. These
findings unde sco e he necessi y o complemen a y expe imen al app oaches
and a p ecise unde s anding o he ansi ions o accu a ely cha ac e ize
magne ocalo ic ma e ials and iden i y sui able candida es o c yogenic
magne ic e ige a ion.
1. In oduc ion
In ecen yea s, he magne ocalo ic effec has become a s ong
con ende in he de elopmen o en i onmen ally iendly
T. Niehoff, E. Byko , C. Salaza Mejía, M. S aßheim, J. Wosni za,
T. Go schall
D esden High Magne ic Field Labo a o y (HLD-EMFL) and
Wü zbu g-D esden Clus e o Excellence c .qma
Helmhol z-Zen um D esden-Rossendo
01328 D esden, Ge many
E-mail: .niehoff@hzd .de
The ORCID iden ifica ion numbe (s) o he au ho (s) o his a icle
can be ound unde h ps://doi.o g/10.1002/ad m.202505704
© 2025 The Au ho (s). Ad anced Func ional Ma e ials published by
Wiley-VCH GmbH. This is an open access a icle unde he e ms o he
C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion
and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly
ci ed.
DOI: 10.1002/ad m.202505704
cooling echnologies a oom
empe a u e.[1–4]Inc easing a en ion
has also u ned o i s applica ions a
c yogenic empe a u es, whe e he MCE
holds significan po en ial o hyd ogen
lique ac ion[5–9]and helium- ee cooling
o c yos a s,[10]offe ing a mo e efficien
cooling al e na i e o exis ing me hods. As
a esul , esea che s a e inc easingly explo -
ing new ma e ials o e isi ing known ones
ha exhibi a s ong MCE a low empe a-
u es. Pa icula ly appealing a e ma e ials
ha combine a p onounced in e se MCE a
e y low empe a u es wi h a con en ional
MCE a highe empe a u es.[11,12]Such
ma e ials can co e he en i e empe a u e
ange equi ed o gas lique ac ion, om
liquid ni ogen a app oxima ely 77 K o
liquid hyd ogen and helium down o 4 K.
P e ious s udies ha e epo ed ma-
e ials exhibi ing his beha io ei he
du ing a me amagne ic ansi ion om
an an i e omagne ic (AFM) o a ully
pola ized e omagne ic (FM) phase[13]
o du ing a ansi ion om a pa amag-
ne ic (PM) o a FM phase. These s udies
p edominan ly ely on magne iza ion measu emen s o indi-
ec ly de e mine he MCE. The s ong in e se MCE obse ed
in such cases is o en a ibu ed o me amagne ic AFM-FM
ansi ions,[13–42]linked o hys e esis,[43]explained h ough
T. Niehoff, M. S aßheim, J. Wosni za
Ins i u ü Fes kö pe - und Ma e ialphysik
Technische Uni e si ä D esden
01069 D esden, Ge many
B. Beckmann, K. Skoko , O. Gu fleisch
Ins i u e o Ma e ials Science
Technical Uni e si y o Da ms ad
Pe e -G ünbe g-S . 16, 64287 Da ms ad , Ge many
A. He e o, A. Oleaga
Depa amen o de Física Aplicada
Escuela de Ingenie ía de Bilbao
Uni e sidad del País Vasco UPV/EHU
Bilbao 48013, Spain
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phenomena such as domain-wall mo emen ,[44]ki-
ne ic a es ,[45,46]cha ge-o de ansi ion,[47]and o he
mechanisms.[48,49]Howe e , hese s udies did no alida e hei
findings using complemen a y echniques, such as specific-hea
measu emen s o di ec de e mina ions o he MCE in pulsed
magne ic fields.[50–53]
In his wo k, we demons a e ha he in e se MCE a low
empe a u es, as p edic ed by magne iza ion measu emen s us-
ing an inco ec applica ion o he Maxwell ela ion, does no
exis . Using a simple model, we e eal he o igin o his appa -
en effec and p o ide a comp ehensi e compa ison o esul s
o Tb3Ni using a ious measu emen echniques. This ma e ial,
p e iously iden ified by some o he au ho s as an example o
his beha io ,[13]se es as a compelling case s udy.
Magne iza ion da a, combined wi h he Maxwell ela ion,[54]
p edic a s ong in e se MCE o Tb3Ni. Howe e , specific-hea
da a e eal no e idence o his effec . Di ec pulsed-field mea-
su emen s, ins ead, e en show subs an ial i e e sible hea ing,
simila o e . [55].
The s onges e idence o he absence o he in e se MCE
comes om he calcula ed o al en opy, as his would iola e he
second law o he modynamics.
Fu he mo e, we gain addi ional insigh in o he ma e ial and
i s ha d magne ic p ope ies h ough simul aneous measu e-
men s o magne iza ion, s ain, and hea flow. These simul ane-
ous measu emen s p o e o be a powe ul echnique o unde -
s anding he p ope ies o he c ys al in de ail.
As men ioned, nume ous pape s ha e been published claim-
ing an in e se magne ocalo ic effec o a ious ma e ials a low
empe a u es, which would hold g ea p omise o c yogenic ap-
plica ions such as gas lique ac ion. Howe e , his effec is, in e-
ali y, absen . Such inco ec p edic ions could ha e se e e con-
sequences, as esea che s would ocus on he w ong ma e ials.
I would only become e iden du ing p ac ical applica ion ha
hese ma e ials a e no sui able.
This pa allels he colossal magne ocalo ic effec , which
was fi s p edic ed in 2004.[56]Se e al highly ci ed wo ks
ollowed,[57–61]p edic ing an ex emely po en magne ocalo ic
effec ha exceeded he heo e ical limi . I was no un il
2009 ha i was e ealed ha his effec came om inco ec
calcula ions.[54,62]
Ou findings, he e o e, highligh he impo ance o employ-
ing complemen a y me hods o accu a ely cha ac e ize magne-
ocalo ic effec s.
2. Resul s and Discussion
2.1. Phase Diag am
To p o ide a clea o e iew and ensu e be e unde s anding o
he p esen ed measu emen s, Figu e 1shows he H-T phase di-
ag am o Tb3Ni o fields applied along he caxis. To cons uc
he phase diag am, we iden ified he phase ansi ions h ough
specific-hea measu emen s as well as by empe a u e- and field-
dependen magne iza ion measu emen s pe o med using a i-
ous p o ocols. The phase diag am aligns closely wi h a p e iously
epo ed diag am.[63]Howe e , we we e unable o esol e he
lock-in ansi ion o he incommensu a e phase wi h ou mea-
020406080100
0
2
4
6
8
10
12 C(T)
M(T)
M(H) DP
M(H) CSP
Pulsed ield
μ
0
H[T]
T[K]
AFM IC
SRO
AFM
FM
Tb
3
Ni
Hc
Figu e 1. H-T phase diag am o Tb3Ni o field applied along he caxis.
The diag am highligh s dis inc magne ic phases, including an i e omag-
ne ic (AFM), e omagne ic (FM), pa amagne ic-like sho - ange-o de ed
an i e omagne ic (SRO AFM), and incommensu a e (IC) an i e omag-
ne ic egions, as de e mined om specific hea Cand magne iza ion mea-
su emen s Munde a ying empe a u e Tand magne ic field Husing a
discon inuous p o ocol (DP) and a con inuous-sweep p o ocol (CSP). The
c ossha ched a ea ep esen s a mixed FM+AFM phase, de e mined by he
hys e esis shown in Figu e 3b. The poin s wi hin he mixed phase we e
iden ified by he addi ional s ep obse ed in he magne iza ion cu e ( o
mo e de ails see e . [63]). The an icipa ed lock-in o he incommensu a e
phase ansi ion[63]is ep esen ed by a pink dashed line a 51 K.
su emen s, which is expec ed o occu a 51 K. The an icipa ed
ansi ion is indica ed as a dashed line in Figu e 1. The ollow-
ing sec ions offe de ailed discussion o he indi idual measu e-
men s and he associa ed phases.
2.2. Specific Hea
Figu e 2ap esen s he specific-hea da a. Se e al ea u es align
well wi h he phase diag am p e iously epo ed by Gubkin
e al.[63]The anomaly a T ,a ound46Ka H=0, was iden i-
fied h ough neu on sca e ing[63]as a lock-in ansi ion in o he
AFM s a e. This ea u e is isible du ing he empe a u e down
sweep, bu absen du ing he up sweep, and i shi s o lowe em-
pe a u es as he magne ic field inc eases [le inse in Figu e 2a].
No ably, his anomaly disappea s en i ely in 3 T.
A ze o field, he specific hea a TN1nea 57 K displays a
b oad and ounded lambda-like anomaly, ma king he ansi ion
om an incommensu a e an i e omagne ic (IC AFM) phase o a
sho - ange o de ed an i e omagne ic phase (SRO AFM). As he
field inc eases o 3 T, his ansi ion is p og essi ely supp essed
and shi s o sligh ly lowe empe a u es. A 4 T, his anomaly is
no longe obse ed.
A 3 T and highe fields, a new 𝛿-peak-like anomaly eme ges a
TC(abou 53 K a 3 T), indica ing a fi s -o de phase ansi ion.
Wi h inc easing field, his peak b oadens and shi s o highe
empe a u es. Beyond 3 T, Tb3Ni has a ansi ion om a low-
empe a u e FM o a PM s a e. A 3 T, he sys em unde goes a se-
quence o phase ansi ions wi h inc easing empe a u es. Fi s
om he FM phase o he IC AFM phase, ollowed by a ansi ion
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Figu e 2. a) Specific hea o Tb3Ni as a unc ion o empe a u e in magne ic fields up o 10 T, o ze o-field-cooled (ZFC) condi ion. The anomalies a T
=46 K and TN1=57 K in ze o field co espond o ansi ions om he an i e omagne ic (AFM) o incommensu a e an i e omagne ic (IC AFM) and
sho - ange o de an i e omagne ic (SRO AFM) phase, espec i ely. TC=53 K a 3 T co esponds o he e omagne ic (FM) o pa amagne ic (PM)
ansi ion, aligning well wi h he phase diag am by Gubkin e al.[63]The le inse highligh s he field-induced shi o T . Addi ionally, he inse shows
ha upon cooling, his ea u e anishes. The igh inse displays a Scho ky-like inc ease. The ea u e a TN2indica es a possible ansi ion o a PM
s a e. The dashed line indica es he Dulong-Pe i alue. b) Tempe a u e-dependen magne iza ion o Tb3Ni o applied field up o 10 T along he easy
caxis o field-cooled (FC) and ZFC condi ions. The FC da a e eal ansi ions co esponding o T ,TN1,TN2,andTC, which align wi h specific-hea
anomalies. The ZFC magne iza ion shows an addi ional ansi ion a fields be ween 2.25 and 4 T and below 25K, disappea ing a 6 T as he sample
becomes e omagne ic. This ea u e is absen in he specific-hea da a. The dashed lines ma k he empe a u es ha a e examined la e (in Figu e 6).
o he SRO AFM phase and finally a c osso e o a PM phase as
he empe a u e inc eases.
Gubkin e al. mapped he phase diag am up o 80 K.[63]How-
e e , ex ending measu emen s o highe empe a u es e eals ad-
di ional ea u es. No ably, a app oxima ely 95 K, a weak anomaly
appea s a TN2in ze o field, which pe sis s up o 4 T wi hou shi -
ing in empe a u e. This ea u e could co espond o a ansi ion
om he SRO AFM phase o a PM s a e.
Addi ionally, we obse e a Scho ky-like inc ease a empe a-
u es below 2 K [ igh inse in Figu e 2a]. Measu ing he hea
capaci y a low empe a u es is c i ical o accu a ely de e min-
ing he en opy change and, wi h his, he magne ocalo ic effec .
Unaccoun ed low- empe a u e anomalies can lead o offse s in
en opy calcula ions, esul ing in subs an ial de ia ions om he
co ec alues. The e o e, he hea capaci y was ex apola ed o
0 K using a polynomial fi in he ange om 0.4 o 1 K o calcu-
la e he en opy. Fo be e cla i y, we summa ize he ansi ions
in he phase diag am shown in Figu e 1.
2.3. Magne iza ion
Figu e 2b shows he empe a u e-dependen magne iza ion o
Tb3Ni, measu ed in fields up o 10 T applied along c, o bo h
FC and ZFC condi ions. Magne iza ion da a up o 1.5 T was p e-
iously epo ed in Re s. [13,64]. The ea u es obse ed in hose
lowe fields, along wi h he anomalies ound he e in he FC mag-
ne iza ion, co espond well wi h he anomalies ex ac ed om
ou specific-hea da a (Figu e 2a). These include he ansi ions
a T ,TN1,TN2,andTC, iden ified by maxima, minima, o slope
changes o he cu es.
The ZFC magne iza ion da a e eal addi ional complexi y. A
fields be ween 2.25 and 4 T, we obse e a sha p inc ease in mag-
ne iza ion below 25 K, indica ing an addi ional me as able phase.
This ea u e shi s o lowe empe a u es as he applied field in-
c eases, and disappea s a 6 T, wi h he sample en e ing a field-
induced e omagne ic s a e a leas abo e 2 K. In e es ingly, his
ansi ion om an an i e omagne ic o a e omagne ic phase is
no isible in he specific-hea measu emen s, nei he in he ZFC
no in he FC da a.
To be e unde s and his ea u e and he magne ocalo ic p op-
e ies, we conduc ed field-dependen magne iza ion measu e-
men s. The p o ocol used is c ucial in ensu ing ep oducible e-
sul s. Figu e 3ashows an illus a ion o he discon inuous p o-
ocol we ollowed (uppe panel), along wi h he co esponding
field-dependen magne iza ion da a (lowe panel). This p o o-
col is ecommended in Re s. [54,65] o ensu e ha he en opy
change calcula ed using he Maxwell ela ion is eliable and no
o e es ima ed due o mixed-phase s a es wi hin a hys e esis loop
o a fi s -o de phase ansi ion. E en when using he discon in-
uous p o ocol, we obse e he me amagne ic ansi ion, clea ly
ma ked by a sha p jump in he magne iza ion.
Figu e 3b shows he magne iza ion o Tb3Ni be ween 14 T and
−14 T a diffe en empe a u es using con inuous field sweeps.
These da a exhibi significan ly diffe en beha io compa ed o
he magne iza ion measu ed using he discon inuous p o ocol
and we e epo ed p e iously by Gubkin e al.[63]Below 20 K,
Tb3Ni u ns in o a ha d magne wi h a coe ci e field o up o 1.5 T
and a emanence close o he sa u a ion magne iza ion. Du ing
his p ocess, he ma e ial ansi ions om a FM o a can ed mag-
ne ic s uc u e and hen back o he FM s a e, as indica ed by he
magne os ic ion da a (see Figu e 6).
The magne iza ion alues ob ained using empe a u e sweeps
(Figu e 2b) and con inuous-sweep p o ocols align well a high
fields. Howe e , he alues measu ed using he discon inuous
p o ocol a e lowe . This disc epancy likely a ises om he ac
ha hese measu emen s we e conduc ed on diffe en pieces
o he sample. While Gubkin e al. eco ded a magne iza ion
o app oxima ely 8.2μB/Tb(μBis he Boh magne on) a 7 T,
ou measu emen s yield 7.37μB/ Tb (230.5Am
2kg−1) using he
con inuous-sweep p o ocol. This de ia ion may esul om ac-
o s such as misalignmen o he sample, o winning o he
c ys al in ou measu emen s. Addi ionally, we obse e a weak
hys e esis be ween 9 T and 12 T ha diminishes o inc easing
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empe a u es (see inse in Figu e 3b). A 14 T, beyond he hys e e-
sis, we ob ained a magne iza ion o 8.06μB/ Tb (252 A m2kg−1).
This is lowe han he heo e ical maximum o gJ =9μBpe Te -
bium a om.
This indica es ha s ong magne oc ys alline aniso opy p e-
en s ull spin alignmen along he magne ic field un il exceeding
12 T, a which poin he comple e spin-pola ized phase is eached.
The s ong aniso opy can be seen in Figu e S4 (Suppo ing
In o ma ion).
2.4. Magne ocalo ic Effec
To quan i y he magne ocalo ic effec , we calcula ed he iso he -
mal en opy change ΔST om he measu emen s desc ibed
abo e. Using he specific-hea da a, we de i ed he field- and
empe a u e-dependen en opy S(T,H) acco ding o he ela ion
S(T, H)=∫T
0(C∕T)HdT. Fo field changes ΔH=H −Hi, he
iso he mal en opy change ΔST(T,ΔH) and he adiaba ic em-
pe a u e change ΔTad(T,ΔH)a egi enby:
ΔST(T, ΔH)=S(T, H )−S(T, Hi)(1)
ΔTad(T, ΔH)=T(S, H )−T(S, Hi)(2)
Addi ionally, we compu ed he iso he mal en opy change us-
ing bo h isofield and iso he mal magne iza ion cu es. In he la -
e case, we ea ed he ZFC and FC cu es sepa a ely. Fo he
calcula ions we used he Maxwell ela ion:
ΔST(T, ΔH)=μ
0∫H
Hi
(𝜕M
𝜕T)HdH (3)
Figu e 4asumma izes hese calcula ed cu es. Fo cla i y,
we p esen only he en opy change ΔST o a field a ia ion
om 0 o 5 T. A empe a u es abo e 40 K, all calcula ed esul s
show excellen ag eemen , e ealing a minimum alue o ΔST
=−17.5 Jkg−1K−1a ound 60 K. This minimum co esponds o
he expec ed con en ional magne ocalo ic effec associa ed wi h
he ansi ion om a pa amagne ic-like SRO AFM phase o he
FM phase.
In con as , below 40 K, s iking disc epancies eme ge be-
ween he esul s ob ained om specific-hea and FC magne iza-
ion da a and hose de i ed om ZFC and field-dependen mag-
ne iza ion da a. The la e wo esul s show a p onounced posi i e
en opy change, indica ing an in e se magne ocalo ic effec . No-
ably, he calcula ion using he ZFC magne iza ion da a exhibi s
a maximum nea 6.3 K, wi h a ema kably la ge en opy change
o ΔST=69 Jkg−1K−1.
This effec o igina es om he anomaly obse ed in he ZFC
and field-dependen magne iza ion da a below 25 K and abo e
2.25 T. This ea u e is caused by he field-induced me amagne ic
ansi ion om he AFM o FM phase, which does no appea in
he specific-hea and FC magne iza ion da a. Con a y o he p e-
ious claim made by some o he au ho s in e . [13] we a gue ha
he calcula ed in e se magne ocalo ic effec is no eal and ha
no addi ional magne ocalo ic effec is p esen .
We show a compelling e idence ha his effec canno be eal
in Figu e 4b, which displays he o al en opy S−S0as a unc-
ion o empe a u e. To de e mine his, we ollowed he s an-
da d app oach o calcula ing he en opy diag am by de i ing he
ze o-field en opy om specific-hea da a and adding he en opy
change o he esul . Fo his, we used he 0 T cu e shown in
Figu e 2a and calcula ed he en opy using:
S(T, H =0) =∫T
0
CH=0
TdT (4)
To ob ain he o al en opy, we added he ΔST(T,ΔH) al-
ues ob ained om he ZFC magne iza ion da a o diffe en
field changes o S(T,H=0). Typically, he expec ed ΔTad alues
a e de i ed om such a diag am. Howe e , o a field change
la ge han 3 T, he en opy dec eases wi h inc easing empe -
a u e abo e 6 K, di ec ly iola ing he second law o he mody-
namics. Acco ding o his undamen al p inciple, inc easing he
he mal ene gy should inc ease he numbe o accessible mic o
s a es, no educe hem.
Figu e 3. a) Magne iza ion da a measu ed using he discon inuous p o ocol (DP) (illus a ed in he uppe panel), highligh ing sha p jumps ha ma k
me amagne ic ansi ions. b) Con inuous field sweeps (CSP) (p o ocol ske ched in he uppe panel) o Tb3Ni om 14 T o −14 T and back, e ealing
la ge hys e esis below 20 K, whe e he compound is a ha d magne . A hys e esis ea u e nea 11 T is highligh ed in he inse .
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Figu e 4. a) Iso he mal en opy change ΔSTcalcula ed om specific-hea , empe a u e-dependen ZFC, FC, and field-dependen magne iza ion da a,
measu ed using he discon inuous p o ocol (DP) o a field change om 0 o 5 T. b) En opy diag am cons uc ed om magne iza ion and ze o-field
specific-hea da a. Fo field changes exceeding 3 T, he en opy cu e dec eases wi h ising empe a u e, con adic ing he second law o he modynamics.
In conclusion, despi e using he usual p o ocol o measu e
he magne iza ion, he in e se magne ocalo ic effec , as de i ed
om he Maxwell ela ion and he ZFC da a, does no exis . In
con as , he en opy change ob ained om he Maxwell ela ion
and FC da a shows good ag eemen wi h ha de i ed om hea -
capaci y da a.
2.5. Adiaba ic Tempe a u e Change
To suppo ou claim, we expanded he s udy by di ec ly measu -
ing he magne ocalo ic effec . In hese measu emen s, we de e -
mined he adiaba ic empe a u e change a he maximum applied
field, ΔTad, as a unc ion o he ini ial sample empe a u e, TS a ,
o fields up o 50 T. We show he da a as ci cles in Figu e 5.Fo
compa ison, we included he adiaba ic empe a u e changes indi-
ec ly de e mined om specific-hea da a ( ha do no p edic an
Figu e 5. Maximum adiaba ic empe a u e change (ΔTad) as a unc ion
o ini ial sample empe a u e (TS a ) in pulsed magne ic fields up o 50 T.
Ci cles ep esen he measu ed ΔTad. Solid lines (pink and blue) indica e
adiaba ic empe a u e changes indi ec ly de e mined om specific-hea
measu emen s o fields o 5 and 10 T. The g ey solid line indica e he dis-
sipa i e hea ing. The dashed lines depic he co ec ed pulsed-field da a,
ma ching he indi ec hea -capaci y esul s and hus also confi ms ha
he e is no in e se calo ic effec .
in e se magne ocalo ic effec ), plo ed as solid lines, calcula ed
using Equa ion (2) o fields o 5 and 10 T.
A empe a u es abo e 40 K, he di ec and indi ec measu e-
men s align well, bu conside able de ia ions appea a lowe
empe a u es. He e, he di ec ΔTad measu emen s e eal a p o-
nounced addi ional effec , simila o he disc epancies obse ed
in he en opy changes shown in Figu e 4a. The key diffe ence is
he e e sed sign o he magne ocalo ic effec . In he di ec mea-
su emen s, we obse e hea ing, which con adic s he expec ed
cooling, associa ed wi h an in e se magne ocalo ic effec .
We p e iously iden ified his i e e sible hea ing in pulsed-
field measu emen s o o he ma e ials as a dissipa i e effec asso-
cia ed wi h a fi s -o de phase ansi ion wi h hys e esis.[55]Du -
ing he field-induced ansi ion om he AFM o he FM phase,
ab up hea ing occu s when he phase bounda y is c ossed. We
ha e quan ified his effec and sub ac ed i om he pulsed-field
da a, yielding co ec ed esul s. These co ec ed pulsed-field da a,
ep esen ed by dashed lines, now closely ma ch he indi ec mea-
su emen s ac oss he en i e empe a u e ange, wi h an absolu e
empe a u e shi , which can be a ibu ed o he diffe en sample
pieces and de ices.
The dissipa i e hea ing can be es ima ed using ΔTdiss =0.5⋅
qdiss ⋅C−1
eff , whe e he dissipa i e ene gy qdiss =μ0∮HdM co e-
sponds o he a ea enclosed by he magne iza ion loop measu ed
du ing he quasi-iso he mal expe imen . Ceff ep esen s he e -
ec i e hea capaci y du ing he apid empe a u e ise. Neglec -
ing he empe a u e dependence can lead o huge unce ain ies,
pa icula ly a low empe a u es. The e o e, o imp o e he es i-
ma e, an effec i e empe a u e is fi s calcula ed as he a e age
empe a u e be ween he s a o he empe a u e ise Tdiss, s a ,
and he final empe a u e eached a e he dissipa i e hea ing,
Tdiss, end. The effec i e a e age empe a u e is Teff =(Tdiss, end +
Tdiss, s a )/2. The effec i e ze o-field hea capaci y Ceff is hen as-
signed based on his empe a u e. The es ima ed ΔTdiss is ep e-
sen ed by a solid g ey line in Figu e 5.
In e es ingly, he dissipa i e effec s a e absen in he low-
empe a u e 5 T pulses a 4.8 and 9.5 K. A hese low empe a-
u es, highe fields a e equi ed o induce he fi s -o de ansi-
ion. This also demons a es ha he dissipa i e effec s a e eal
and canno be a ibu ed o eddy cu en s.
Focusing only on he ΔTad abo e 40 K, we obse e he
s ong po en ial o Tb3Ni o low- empe a u e magne ocalo ic
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Figu e 6. Field dependence o adiaba ically measu ed sample empe a u e Tad, magne iza ion M, s ain, and hea flow ΔTflow a a)5K,b)15K,c)40K,
and d) 60 K. The g een cu es show he simul aneously measu ed Tad and M. The FC magne iza ion, s ain, and hea flow we e measu ed simul aneously
in quasi-s a ic fields.
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e ige a ion. Fo 10 T pulses, he ma e ial exhibi s a maximum
e e sible adiaba ic empe a u e change o ΔTad =10.5 K a a
s a ing empe a u e TS a o 66 K. As he field inc eases, he max-
imum shi s o highe empe a u es, eaching a ΔTad o 33 K a
TS a =163 K o 50 T pulses.
These di ec measu emen s a e c ucial o ully cha ac e -
izing Tb3Ni, as hey conclusi ely demons a e ha he in-
e se magne ocalo ic effec sugges ed by he indi ec calcu-
la ion is no eal. Fo an e en deepe unde s anding, we
p esen he field-dependen da a o he indi idual pulses in
Figu e 6 o a ious s a ing empe a u es. These da a illus-
a e he dissipa i e hea ing as well as he magne ocalo ic
effec .
2.6. Simul aneous Measu emen s
Figu e 6p esen s he simul aneous field-dependen measu e-
men s o FC magne iza ion, s ain, and hea flow ΔTflow,aswell
as ZFC magne iza ion, and pulsed-field da a a a ious empe -
a u es. To p o ide be e insigh , we indica ed he co espond-
ing empe a u es as dashed lines in Figu e 2b. We pe o med
he s ain and hea -flow measu emen s alongside he FC mag-
ne iza ion measu emen s shown in Figu e 3b. These measu e-
men s ollowed he con inuous sweep p o ocol, wi h he down-
sweep da a ep esen ed as M FC down and he up-sweep da a as
MFCup.
Addi ionally, we included magne iza ion measu emen s om
he discon inuous p o ocol, labeled as M ZFC. The figu e also
includes he di ec ly measu ed magne ocalo ic effec in pulsed
field, Tad, oge he wi h he simul aneously measu ed magne i-
za ion, labeled M pulsed.
Figu e 6a,b illus a e he di ec ansi ion om he AFM o he
ully pola ized FM phase a 5 and 15 K, espec i ely. This fi s -
o de phase ansi ion shows p onounced hys e esis, which we
clea ly obse e in he magne iza ion, as well as in he adiaba ic
empe a u e changes du ing pulsed-field measu emen s. Dissi-
pa i e hea ing, caused by he hys e esis, gene a es a sha p em-
pe a u e jump as he ansi ion occu s depic ed in he op panel
o Figu e 6a,b. A 5 K, his jump occu s nea 5.5 T, while a 15 K,
i appea s a ound 4.5 T. A e he ansi ion, bo h samples hea o
app oxima ely 25 K. This co esponds o he empe a u e, abo e
which no hys e esis occu s. Fo his eason, no u he dissipa i e
hea ing can ake place. The empe a u e eached h ough dissi-
pa i e hea ing is consis en ly 25 K. Below his empe a u e, ΔTad,
asshowninFigu e5, only a ies due o he diffe en ini ial em-
pe a u es. Du ing he down sweep, we obse e a second small
empe a u e inc ease a ound 2 T as he sample een e s he AFM
phase. This esul s om he na ow hys e esis nea 2 T a 25 K
(see Figu e 3b). The empe a u e inc ease is u he educed by
he sample’s now la ge hea capaci y a 25 K. Beyond hese dis-
sipa i e effec s, we de ec no addi ional field-dependen empe -
a u e changes.
The op panels o Figu e 6c,d (40 and 60 K, espec i ely)
demons a e he ypical beha io o a e e sible magne ocalo ic
effec du ing a fi s -o de phase ansi ion. A he ansi ion, he
empe a u e Tad ises quickly and con inues o inc ease as he
sample is in he FM phase. Du ing he down sweep, he empe -
a u e dec eases cong uen ly wi h he up-sweep. When he phase
ansi ion is eached again, wi h a hys e esis o app oxima ely 1 T,
he empe a u e e u ns o i s ini ial alue. These esul s confi m
he p esence o a e e sible magne ocalo ic effec a highe em-
pe a u es du ing bo h he IC-FM and SRO AFM-FM ansi ions,
while only dissipa i e hea ing is p esen a lowe empe a u es
du ing he AFM-FM ansi ion.
I is also no iceable, ha he empe a u e jump in pulsed-
field measu emen s appea s abou 2 T abo e he phase ansi-
ion ex ac ed om quasi-s a ic measu emen s. This shi could
esul om dynamic effec s du ing he apid pulsed-field mea-
su emen . Highligh ed in he phase diag am (Figu e 1), his shi
occu s only du ing he AFM- o-FM ansi ion and disappea s a
highe empe a u es, when he s a ing empe a u e lies wi hin
he IC o SRO AFM phase.
Addi ionally, we obse e ha a 5 K Figu e 6a), he con inu-
ous sweep p o ocol leads o a lowe field o induce he FM an-
si ion compa ed o he discon inuous p o ocol. This diffe ence
anishes a highe empe a u es, whe e he sample e ains i s
ha d magne ic p ope ies bu no longe exhibi s a mixed s a e,
as e idenced by he in e media e s ep on he FC magne iza ion
hys e esis da a. This ea u e is e iden in bo h s ain and hea -
flow measu emen s. When he coe ci e field o 1 T is eached,
he change in magne iza ion is accompanied by a empe a u e
inc ease and a longi udinal elonga ion along he applied field o
app oxima ely 0.05 %, while he ans e se s ain only shows a
small change o abou −0.01 %. A 2 T, we obse e an addi ional
kink in he in e media e ansi ion in bo h he s ain and mag-
ne iza ion measu emen , bu no in he hea -flow da a. A 3.5 T,
as he sample ully ansi ions in o he FM phase, he empe a-
u e ises again, and he s ain e u ns o i s ini ial s a e. Beyond
his poin , he s ain inc eases s eadily in he FM phase, which
appea s as well a highe empe a u es, bu wi h a s onge in-
c ease in s ain. In he AFM phase, he s ain emains cons an ,
independen o empe a u e. A 15 K, he ea u es a he AFM- o-
FM ansi ion diffe s om Figu e 6a because he sample is no
longe a ha d magne , leading o dis inc s ain da a. Du ing he
ansi ion, he sample comp esses along he field di ec ion by ap-
p oxima ely 0.05 %, simila o he beha io a 5 K, bu addi ionally
his comp ession is coun e ac ed by expansion in he ans e se
di ec ion o also abou 0.05 %. A simila beha io appea s a 40 K
(Figu e 6c).
A 60 K (Figu e 6d), he s ain beha io changes again as Tb3Ni
unde goes he IC- o-FM phase ansi ion. In he IC phase, he
sample con ac s sligh ly wi h inc easing field. As i ansi ions
o he FM phase, bo h longi udinal and ans e se s ain inc ease,
esul ing in a olume ic expansion.
2.7. Mean-Field Model
To explain why indi ec calcula ions o he magne ocalo ic effec
om magne iza ion measu emen s may inco ec ly p edic an
in e se magne ocalo ic effec , and o demons a e ha his dis-
c epancy is no unique o Tb3Ni, we analyze a simplified model
sys em. This example highligh s inconsis encies depending on
he me hod used o de e mine he magne ocalo ic effec .
The model unde conside a ion ea u es an i e omagne ic
in e ac ions be ween nea es -neighbo a oms, compe ing wi h
e omagne ic couplings be ween nex -nea es neighbo s. Fo
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Figu e 7. a) Calcula ed no malized magne iza ion mas a unc ion o educed empe a u e o diffe en educed fields o h=0.5, 0.8, 0.9, 0.98, 1, 2,
and 4 using Equa ion (5)wi h|J′/J|=1. The ab up magne iza ion jumps o he AFM-FM ansi ion a e indica ed by dashed lines. b) En opy change Δs
as a unc ion o educed empe a u e, compa ing calcula ions based on he ee ene gy wi h hose de i ed indi ec ly using he Maxwell ela ion.
e e ence, measu emen s on Tb3Ni ha e es ablished ha a
ze o field and low empe a u es, an i e omagne ic coupling
domina es.[13,63]Addi ionally, he sys em is ep esen ed by an
Ising model, whe e spins can only align pa allel o an ipa allel o
he caxis. This beha io mi o s he s ong aniso opy obse ed
in Tb3Ni be ween easy and ha d axis[63](see Figu e S4, Suppo -
ing In o ma ion). In con as , a Heisenbe g model, which allows
spins o o ien iso opically, would no ep oduce he obse ed
ab up magne iza ion jump du ing he AFM-FM ansi ion, and
may e en show spin can ing.[66]The Hamil onian o his sim-
plified sys em is:
=−J∑
⟨i,j⟩
Si⋅Sj−J′∑
⟨⟨i,k⟩⟩
Si⋅Sk−h∑
i
Si(5)
whe e Si=±1 ep esen s an Ising spin a si e i. The fi s e m
desc ibes an i e omagne ic in e ac ions be ween nea es neigh-
bo s (J<0). The second e m accoun s o e omagne ic in e -
ac ions be ween nex -nea es neighbo s (J′>0). The hi d e m
in oduces an ex e nal magne ic field (h).
In he pas , esea ch has p ima ily ocused on e omagne ic
models o fi s - and second-o de ansi ions,[67,68]wi h some
s udies also in es iga ing an i e omagne ic sys ems.[69]Mod-
els ha conside bo h an i e omagne ic and e omagne ic cou-
plings ha e mos ly been analyzed using Heisenbe g models.[70,71]
O he app oaches include mul ilaye sys ems wi h an i e omag-
ne ic in e plana and e omagne ic in aplana couplings.[72]
Ising models ha e also been in es iga ed using Mon e Ca lo
me hods, bu wi h s ong e omagne ic coupling |J′/J|>50[73]
o in pe o ski e s uc u es wi h |J′/J|=1.[74]Howe e , nei he
Heisenbe g no he men ioned Ising models ha e been able o
ep oduce he ab up inc ease in magne iza ion du ing he me a-
magne ic ansi ion. A simila Ising model ha is compa able o
he one in es iga ed he e has been s udied in e . [75]. E en in
his case, he ab up me amagne ic ansi ion was no examined
in de ail.
We sol e he model Equa ion (5) using he B agg–Williams
mean-field app oxima ion, desc ibed in p e ious wo ks.[66,76–78]
Fo simplici y, we use educed a iables, wi h he empe a u e
=kBT/(|J+J′|), he Bol zmann cons an kB, and he field h=
μ0H/(|J+J′|). Fu he de ails can be ound in he Suppo ing In-
o ma ion.
Fo s ong e omagne ic coupling (|J′/J|>2), he sys em no
longe shows an an i e omagne ic phase and o weak FM cou-
pling (|J′/J|<0.2), he ansi ion be ween AFM and FM s a es has
no ab up magne iza ion jump.
Fo his s udy, we ocus on a coupling a io o |J′/J|=1,
whe e he compe i ion be ween he in e ac ions p oduces he
desi ed ea u es. Addi ionally, we calcula ed he magne iza ion
jump om he hys e esis o he model by minimizing he ee
ene gy. We p esen he no malized magne iza ion as a unc ion
o empe a u e in Figu e 7a, wi h he magne iza ion jumps indi-
ca ed by dashed lines. Despi e he simplici y o he model, he
magne iza ion closely esembles he empe a u e dependence
obse ed o Tb3Ni (see Figu e 2b). A low fields, he magne i-
za ion beha es as expec ed o an an i e omagne up o h=0.8.
As he field inc eases, he magne iza ion ab up ly ansi ions o
a ully pola ized e omagne ic s a e a lowe empe a u es. The
size o he jump g ows wi h field and occu s a p og essi ely
lowe empe a u es. Abo e h=1.0, he an i e omagne ic s a e
no longe exis s a he lowes empe a u es, and he model be-
ha es like a e omagne . The model also p edic s s ong hys e e-
sis du ing he ab up AFM-FM ansi ion, which aligns well wi h
he beha io obse ed in Tb3Ni (See Figu es S1 and S2, Suppo -
ing In o ma ion). In pa icula , in expe imen , we ound a no-
able dissipa i e hea ing du ing pulsed-field measu emen s, ha
only occu s when he AFM-FM ansi ion is c ossed. Howe e ,
he simple model does no accoun o he ha d magne ic p op-
e ies exhibi ed by Tb3Ni (which would equi e a mo e sophis i-
ca ed heo y).
In Figu e 7b, we p esen bo h he en opy change Δs di ec ly
calcula ed wi h he ee ene gy ( ,h) ob ained using Equa ion (5)
( u he de ails can be ound in he Sec ion S1, Suppo ing In-
o ma ion.), s( , h)=−
𝛿 ( ,h)
𝛿 , and Equa ion (1) as well as he indi-
ec ly de e mined alue using he Maxwell ela ion:
Δs ( , h )=∫hc
0(𝜕m
𝜕 )hdh +𝛿s( , hc)
+∫h
hc
(𝜕m
𝜕 )hdh
(6)
The fi s in eg al accoun s o he egion up o he AFM-
FM ansi ion a he c i ical field hc, while he second in eg al
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co e s he egion beyond he ansi ion. The en opy jump
𝛿s( ,hc) a he ansi ion can be es ima ed using he Clausius-
Clapey on equa ion.[79]
A low fields (h=0.5), he sys em exhibi s bo h an in e se and
a con en ional magne ocalo ic effec a ound he Néel empe a-
u e, esul ing in posi i e and nega i e Δs , espec i ely. This be-
ha io is expec ed o an AFM ansi ion[71,74,80]and can also be
obse ed a low fields in Tb3Ni, as shown in Figu e S3 (Suppo -
ing In o ma ion). A highe fields (h=2), he calcula ions based
on he ee ene gy esul s in a con en ional magne ocalo ic e -
ec , cha ac e ized by a nega i e en opy change, which is con-
sis en wi h a e omagne ic ansi ion. Howe e , when he en-
opy change a h=2 (a e he magne iza ion jump), is calcu-
la ed indi ec ly using he Maxwell ela ion Equa ion (6), a pos-
i i e en opy change eme ges a empe a u es below =1.3
p edic ing an in e se magne ocalo ic effec . This addi ional effec
is he esul o he addi ional Clausius-Clapey on e m in Equa-
ion (6) and he discon inui ies in he magne iza ion da a. In eal
measu emen s, hese a e no magne iza ion jumps bu a he di -
e en iable unc ions. The e o e, he simple Maxwell ela ion is
used in expe imen s, which would also lead o a non-exis en in-
e se magne ocalo ic con ibu ion. A highe empe a u es, he
en opy change calcula ed by bo h me hods is iden ical. A low
fields (h=0.5), whe e no magne iza ion jump is obse ed, he
en opy changes coincide ac oss he en i e empe a u e ange o
bo h calcula ion me hods.
The ac ha a genuine in e se magne ocalo ic effec occu s
a low fields due o he an i e omagne ic ansi ion, disappea s
a highe fields, and ye Maxwell’s ela ions p edic an in e se e -
ec o fields abo e he me amagne ic ansi ion is p obably he
eason why his effec has o en been misin e p e ed in he li -
e a u e, as i has been inco ec ly a ibu ed o he an i e omag-
ne ic ansi ion.
In summa y, he en opy change calcula ed om he ee en-
e gy accu a ely p edic he ac ual beha io , aligning wi h he mag-
ne ocalo ic effec de e mined om hea -capaci y measu emen s
o Tb3Ni. In con as , he in e se magne ocalo ic effec sugges ed
a high fields and low empe a u es by he Maxwell ela ion is in-
co ec , bo h in his simplified model and in expe imen , because
o he fi s o de ansi ion.
To u he cla i y he diffe ence be ween he absence o he in-
e se magne ocalo ic effec and he dissipa i e hea ing, Sec ion 4
(Suppo ing In o ma ion) p esen s he calcula ed Δ and he dis-
sipa i e hea ing.
We emphasize ha , while a mo e complex Hamil onian, in-
cluding addi ional couplings, would be e cap u e he beha io
o Tb3Ni, his simplified model can demons a e he essen ial
physics. Specifically, ha he coexis ence o compe ing AFM and
FM in e ac ions, combined wi h s ong aniso opy, is sufficien o
p oduce a magne iza ion jump accompanied by significan hys-
e esis. No ably, his does no esul in an in e se MCE when cal-
cula ed om he ee ene gy, highligh ing how he Maxwell ela-
ion can yield inco ec esul s
3. Conclusion
In his wo k, we demons a e ha de e mining he MCE only
based on magne iza ion measu emen s and he Maxwell ela-
ion, widely used in he li e a u e due o i s simplici y and expe -
imen al accessibili y, can lead o d as ic misin e p e a ions, es-
pecially in ma e ials wi h la ge hys e esis. Despi e using p ope
measu emen p o ocols, many ma e ials ha e been w ongly cha -
ac e ized, wi h o e ly op imis ic p edic ions o p omising effec s
ha we e no alida ed by complemen a y measu emen s.
Using Tb3Ni as a case s udy, we confi m ha he in e se calo ic
effec is no eal. We achie e a comple e unde s anding o he
magne ocalo ic beha io h ough a combina ion o expe imen-
al echniques, including magne iza ion, specific-hea , and di ec
pulsed-field ΔTad measu emen s. While he Maxwell ela ion p e-
dic s a p onounced in e se calo ic effec , specific-hea measu e-
men s e eal no such effec , and pulsed-field ΔTad measu emen s
e eal a s ong con en ional magne ocalo ic effec . The appa -
en in e se calo ic effec is a ibu ed o an imp ope applica ion
o he Maxwell ela ion, and he s ong hea ing obse ed in di-
ec measu emen s is iden ified as a consequence o dissipa i e
effec s.
Fu he mo e, we applied a simple model o illus a e he o igin
o hese misin e p e a ions, demons a ing ha his is no lim-
i ed o Tb3Ni, bu ep esen s a undamen al phenomenon ha
can occu ac oss a wide ange o ma e ials.
Addi ionally, ou findings highligh he alue o combining
simul aneous expe imen al echniques, such as magne iza ion,
s ain, and hea -flow measu emen s, o unco e he complex be-
ha io o ma e ials, p o iding deepe insigh s in o hei in in-
sic p ope ies.
Finally, elying on magne iza ion measu emen s alone is in-
sufficien o accu a ely cha ac e izing he magne ocalo ic e -
ec . To ully unde s and he magne ocalo ic effec o a ma e-
ial, he phase ansi ions and unde lying physics o he sys-
em mus be well unde s ood. A comp ehensi e expe imen al
app oach, inco po a ing mul iple echniques such as magne i-
za ion, hea capaci y, and di ec measu emen s o he magne-
ocalo ic effec , o ins ance, in pulsed fields, is needed o ensu e
he eliabili y o epo ed esul s. Wi hou such alida ion, he
ue po en ial o he magne ocalo ic effec , pa icula ly o eme g-
ing applica ions such as c yogenic cooling, may be misin e -
p e ed and o e s a ed, he eby hinde ing p og ess in his field o
esea ch.
4. Expe imen al Sec ion
Sample P epa a ion:In his wo k, he same Tb3Ni c ys al was used as
desc ibed in Re s. [13,63]. The c ys al was syn hesized by p epa ing a poly-
c ys alline ingo o Tb3Ni h ough a c mel ing in a helium a mosphe e,
wi h e bium o 99.9 % pu i y and nickel o 99.99 % pu i y. Se e al single-
c ys al samples we e g own by emel ing he ingo a empe a u es sligh ly
abo e he pe i ec ic poin in a esis ance u nace, ollowed by slow cooling.
F om hese, a single c ys al measu ing app oxima ely 4 ×5×6m
3m was
ex ac ed. The c ys al quali y was assessed h ough X- ay diff ac ion using
he back- eflec ion Laue me hod on a ious su aces, confi ming ha no
addi ional eflec ions om o he g ains we e p esen . The esul ing sam-
ple was shaped in o a pa allelepiped, wi h su aces o ien ed pe pendicu-
la ly o he c ys allog aphic a,b,andcaxes. A e he su aces we e pol-
ished, hei o ien a ion was once mo e e ified h ough Laue diff ac ion.
Fo magne iza ion and hea -capaci y measu emen s, he sample was cu
in o se e al smalle pieces, each weighing app oxima ely 4 mg.
Specific Hea :The specific hea was measu ed in magne ic fields up o
10 T be ween 400 mK and 150 K. Below 2 K, he hea -pulse me hod was em-
ployed using a 3He c yos a (Ox o d). Fo empe a u es om 2 o 150 K, we
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