1
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La ice dynamics in CePd2Al2
and LaPd2Al2
Pe Doležal1*, Pe Cejpek1, Sa oshi Tsu sui2,3, Koji Kaneko4, Dominik Legu 5, Ka el Ca a1 &
Pa el Ja o ský1
The in e ac ion be ween phonons and 4 elec ons, which is o ming a new quan um s a e (quasi-
bound s a e) beyond Bo n-Oppenheime app oxima ion, is e y p ominen and la ice dynamics plays
he e a key ole. The e is only a small numbe o compounds in which he expe imen al obse a ion
sugges such a scena io. One o hese compounds is CePd2Al2. He e he s udy o phonon dispe sion
cu es o (Ce,La)Pd2Al2 a 1.5, 7.5, 80 and 300 K is p esen ed. The inelas ic X- ay sca e ing echnique
was used o mapping he phonon modes a X and Z poin s as well as in Λ and Δ di ec ions, whe e he
symme y analysis o phonon modes was pe o med. The measu ed spec a a e compa ed wi h he
heo e ical calcula ion, showing e y good ag eemen . The measu emen s we e pe o med in se e al
B illouin zones allowing he econs uc ion o phonon dispe sion cu es. The esul s a e discussed
wi h espec o he magne o-elas ic in e ac ion and a e compa ed wi h o he ce ium compounds. The
phonon mode symme y A1g was ound o be una ec ed by he in e ac ion, which is in con as o
p e ious assump ions.
Physical p ope ies o in e me allic compounds a e d i en by hei elec onic p ope ies and la ice dynamics,
bu usually hei impac is ea ed sepa a ely. This is also he case o magne ism in a e ea h compounds. The
localised cha ac e o 4 elec ons and hei geome ical a angemen in he c ys al la ice de e mine he magne ic
beha iou and he la ice dynamics is usually no conside ed. This app oach is based on he Bo n-Oppenheime
app oxima ion, whe e he di e ence o se e al o de s o magni ude in mass o a omic nucleus and elec ons
ypically jus i y his a i ude.
In s anda d c ys al elec ical ield (CEF) heo y he a e ea h ion wi h ull o a ion symme y is placed in o a
s a ic c ys al la ice, which dec eases deg ees o eedom and educes he degene acy o a g ound s a e depending
on he poin symme y o a gi en si e in he c ys al la ice. I was shown, ha his app oach canno explain he
magne ic exci a ion spec a measu ed in se e al compounds, e.g. in py ochlo es Tb2Ti2O71, Ho2Ti2O72, in e me-
allics CeAl23, P Ni24, CePd2Al25, CeCuAl36 and ecen ly in CeAuAl37. The Ce-based in e me allics ha e a e y
p ominen posi ion wi h a ela i ely small numbe o expec ed CEF exci a ions. The oxida ion numbe o he
Ce ion in he ma e ials lis ed abo e is 3+, implying o al angula momen um
J=
5
2
and a six old degene a ed
g ound s a e. The hal in ege momen equi es a leas wo old degene acy e en in iclinic si e symme y in
he c ys al la ice. This simply es ic s he numbe o magne ic exci a ions om he g ound s a e o wo. The
highe numbe o obse ed magne ic exci a ions in he spec a han wo is e y peculia . One o he possible
explana ions o such beha iou was ound in he coupling o la ice dynamics and 4 elec onic s a es, which is
usually neglec ed in he Bo n-Oppenheime app oxima ion as men ioned abo e. This would lead o he o ma-
ion o a new quan um s a e8 and opens new ques ions abou i s o igin and condi ions which ha e o be ul illed.
All hese aspec s make hese compounds e y in e es ing especially in he case when an X- ay ee elec on lase
opens new possibili ies o s udy he la ice dynamics.
The CeAl2 c ys allises in a cubic s uc u e o ype MgCu2 La es phase9. The si e symme y o he Ce ion is
also cubic, he e o e only wo le els
Ŵ2
7
and
Ŵ4
8
(based on CEF heo y) and one exci a ion om g ound s a e a e
expec ed. Su p isingly, wo exci a ions om he double
Ŵ2
7
ha e been obse ed by inelas ic neu on sca e ing3.
Simul aneously he s udy o phonon dispe sion cu es by inelas ic neu on sca e ing shows a so ening and
b oadening o he
Ŵ′
25
phonon mode and symme y ela ed phonon b anches in he

di ec ion10. Bo h obse ed
ea u es we e explained by he exis ence o a quasi-bound s a e be ween 4 elec ons and phonons8. The in e -
ac ion e m in he model Hamil onian uses he c ys al elec ical ield ope a o wi h he same ans o ma ion
OPEN
1Facul y o Ma hema ics and Physics, Depa men o Condensed Ma e Physics, Cha les Uni e si y, Ke Ka lo u
5, 121 16 P ague 2, Czech Republic. 2Japan Synch o on Radia ion Resea ch Ins i u e (JASRI), SP ing-8, Sayo,
Hyogo 679-5198, Japan. 3Ins i u e o Quan um Beam Science, G adua e School o Science and Enginee ing, Iba aki
Uni e si y, Hi achi, Iba aki 316-8511, Japan. 4Ma e ials Science Resea ch Cen e , Japan A omic Ene gy Agency,
Tokai, Iba aki 319-1195, Japan. 5IT4Inno a ions, VSB-Technical Uni e si y o Os a a, 17. lis opadu 2172/15, 708
00 Os a a, Czech Republic. *email: Pe .Dolezal@ka lo .m .cuni.cz
2
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p ope ies as he
Ŵ′
25
mode8. This shows how he knowledge o phonon dispe sion cu es play a key ole in sea ch
o he model Hamil onian. Wi hou di ec obse a ion o he in luence o a quasi-bound s a e on phonons, he
p esence o he quasi-bound s a e emains s ill an open ques ion.
The expe imen al s udies in CeCuAl36, CePd2Al25 and CeAuAl37 p opose he exis ence o a bound s a e, bu
i s signa u e was ound mainly in he magne ic exci a ions spec a. The s udy o phonon dispe sion cu es is s ill
missing wi h some excep ion o CeAuAl3 and CeCuAl3711. The s udy o low ene gy phonon dispe sion cu es
(mainly acous ic) on CeAuAl37 shows he an ic ossing o a non-dispe si e magne ic exci a ion and an acous ic
phonon b anch, which was no obse ed be o e. The so ening o b oadening as known o CeAl2 was no ound
in he measu ed ene gy in e al7.
Ou p esen s udy o single c ys al CePd2Al2 and LaPd2Al2 is ocused on phonon dispe sion cu es s udied
by inelas ic X- ay sca e ing. The non-magne ic La homologue is used as a e e ence sample, because he La ion
ha e no 4 elec ons and he e o e he bound s a e be ween 4 elec ons and phonons is no p esen he e. We
pe o med he symme y analysis o phonon modes a
Ŵ
and X poin s and in he

and

di ec ions, which we e
calcula ed in12. Based on he symme y analysis we ocused on selec i e phonon modes, especially wi h symme y
used in he model Hamil onian5 and compa e i wi h he expe imen al da a. The esul s a e discussed in e ms
o a possible appea ance o a quasi bound s a e in hese Ce based in e me allics.
Me hods
Single c ys als o CePd2Al2 and LaPd2Al2 we e p epa ed by he Czoch alski me hod. As hese compounds a e
incong uen ly mel ing, he p ocedu e desc ibed in13 was adop ed. The s a ing composi ions o he mel ed p ecu -
so s we e 22:41.1:36.9 (Ce:Pd:Al) and 22:39:39 (La:Pd:Al). The phase pu i y in he p epa ed ingo s was checked
by X- ay powde di ac ion pe o med on B uke D8 Ad ance di ac ome e in B agg-B en ano geome y
wi h a iable sli s. The pa e ns o bo h Ce and La homologs we e i by a CaBe2Ge2 s uc u al model using he
Rie eld me hod wi h esul ing la ice pa ame e s
a=b=4.4112(3)
Å,
c=9.8749(9)
Å and
a=b=4.439(1)
Å,
c=9.915(3)
Å, espec i ely. The chemical composi ion was analysed by Ene gy-dispe si e X- ay analysis and
phase pu i y was also con i med by backsca e ed elec ons imaging pe o med on an elec on mic oscope
TESCAN, ype Mi a I LMH. The esul ing composi ion o samples we e 18:41:41 and 20:40:40 a omic p ecen o
Ce:Pd:Al and La:Pd:Al. The measu ed chemical composi ion co esponds o ideal one wi hin 2% expe imen al
e o . The sample sizes o Ce and La based samples we e 1x0.6x0.6mm and 1x0.8x0.4mm, espec i ely.
The single c ys alline samples we e o ien ed by Laue me hod using a Laue X- ay Imaging Sys em 20041209 SY
Issue 8 (Pho onic Science) wi h CCD came a (1220-1800 pixels) and ai -cooled X- ay ube (30 kV, 300 μA). The
beam size 0.5 mm2 used o Laue me hod was smalle han he sample. A ligh -pola ized mic oscopy was used
o con i m he single-g ain na u e o he samples.
The p e ious s udy o polyc ys alline (Ce,La)Pd2Al2 samples e eals he p esence o a s uc u al phase ansi-
ion o a lowe symme y o ho hombic s uc u e desc ibed by he Cmme(67) space g oup5,14. The empe a u e
o he s uc u al phase ansi ions,
Ts
, in p epa ed single c ys als was checked by low empe a u e X- ay pow-
de di ac ion showing ha he s uc u al ansi ions a e shi ed o lowe empe a u es in compa ison o he
polyc ys alline samples,
Ts
≅
6
K o CePd2Al2 and
Ts
≅
60
K o LaPd2Al2. In polyc ys alline samples he
ansi ion empe a u es we e a ound 13.5K (11K) and 91.5K (84K)5(14) o Ce and La samples, espec i ely.
De ails o he low empe a u e di ac ion a e p esen ed in Supplemen a y in o ma ion.
The inelas ic X- ay sca e ing spec a we e measu ed a he BL35XU beamline a he SP ing-8 synch o on
(Japan). The high o de 111111 di ac ion om a Si c ys al and ene gy o 21.747keV o he beam we e used o
ensu e he ene gy esolu ion o 1.5meV. The beam size was
≈50
μm. Si c ys als we e also used as analyse and
hei la ice pa ame e s we e con olled by a empe a u e, which allows he selec ion o he ene gy ans e . The
de ail desc ip ion o BL35XU is gi en in1516.
Theo e ical calcula ion o phonon dispe sion cu es in (Ce,La)Pd2Al2 a e p o ided in12. Fo he analysis o
inelas ic spec a is necessa y o know he pola isa ion and phase o displacemen o all a oms in he p imi i e
uni cell, which a e gi en by eigen ec o s
eqjd
and we e calcula ed in12. The in ensi y o he selec ed phonon
mode in a gi en B illouin zone is p opo ional o dynamical s uc u e ac o , which one-phonon con ibu ion
is gi en by16:
whe e N coun s he i adia ed uni cells,
τ
is he B agg ec o ,
q
is he educed momen um ans e in he
i s B illouin zone,
Q=q+
τ . The index j uns o e phonon modes o a gi en
q
and d uns o e he a oms
in he p imi i e uni cell. The posi ion o a om wi hin he uni cell is gi en by
xd
,
Md
is i s mass and
d
is he
a omic o m- ac o o X- ays.
Fqj(ω)
co esponds o he ene gy esolu ion unc ion.
Wd
(Q))
=
B
d
||Q||
2
16π
2 is he
Debye-Walle ac o . Fo ou calcula ions, we use he Debye-Walle ac o app oxima ed o he case o iso opic
ib a ions17 as:
whe e T is he empe a u e,
D
is Debye empe a u e and h and
kB
a e Planck and Bol zmann cons an espec-
i ely.
φ
is he Debye unc ion
φ(
x)
=
1
x

x
0
ξ
e
ξ
−1
d
ξ
.
(1)
S
(Q,ω)1p=N
q

j





d
d(Q)
√2Mde−Wd(Q)Q·eqjd eiQ·xd



2
δ(Q−q)τFqj(ω)
,
(2)
B
d
=
6h
2
MdkB�D
φ(�D/T)
�D/T+
1
4

3
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The measu emen was done in hese B illouin zones 0 1 8, 0 0 8, 0 1 12, 0 2 14, 4 0 1, 3 0 1 and 4 0 0. This
selec ion was chosen o co e a wide ange o phonon b anches wi h special a en ion o he possible sepa a ion
o mode in ensi ies du ing he da a analysis based on he pe o med simula ion gi en by Equa ion1.
Resul s – Phonon dispe sion cu es in (Ce,La)Pd2Al2
The ib a ions o a omic nuclei in he c ys al s uc u e a e desc ibed by no mal modes, which o m he pho-
non dispe sion cu es in he B illiouin zone. These modes a e classi ied and labelled acco ding o i educible
ep esen a ions (IRs) o he g oup o
k
ec o (li le g oup)
Gk
con ained in mechanical ep esen a ion. Be o e
inding IR o
Gk
i has o be no ed ha we ha e symmo phic (SSG) and non-symmo phic space g oups (NSG).
In he case o SSG, all ansla ions can be ac o ised, and he ac o g oup
¯
Gk
o m a poin g oup. Unlike SSG,
he NSG con ain glide planes and sc ew axes, and consequen ly he
¯
Gk
is no isomo phic o a poin g oup in
gene al. This means ha only a he
Ŵ
poin ,
k=0
, he decomposi ion in o IRs o poin g oups is possible o
bo h SSG and NSG. In he ollowing ex , he phonon modes and phonon b anches a e labelled acco ding o he
solid s a e no a ion used in18. Only a he
Ŵ
poin , whe e he li le g oup
Gk
is iden ical o he poin g oup, he
no a ion used in chemis y is gi en in b acke s. Away om he cen e o he B illiouin zone, he symme y o
phonon dispe sion cu es we e de e mined by using he compa ibili y ela ions in18.
The poin g oup o he
P4/nmm
space g oup is D
4h
. Fo symme y de e mina ion o he phonon dispe sion
cu es a he
Ŵ
poin , we used he di ec p oduc o equi alence ep esen a ion and ep esen a ion o he pola
ec o :
The e a e he e o e 20 di e en phonon modes (30 wi h he conside a ion o degene acy) a he
Ŵ
poin . This is
in ag eemen wi h he ac ha 10 a oms a e in he p imi i e uni cell o CePd2Al2. All he 30 modes a e lis ed
in he Supplemen a y ma e ial and labelled wi h a numbe (#No.) inc easing wi h hei ene gy. These numbe s
a e used u he in he ex .
The a o emen ioned symme y analysis allows us o sepa a e he calcula ed phonon b anches in12 acco ding
o hei symme y wi hin he B illiouin zone. Figu e1a shows highe symme y poin s and di ec ions in he
p imi i e e agonal uni cell using he no a ion in18. Ou measu emen s we e pe o med in

and

di ec ions.
Ou symme y analysis, shown in Fig.2, is he e o e limi ed o hese di ec ions. The ou panels highligh always
one symme y ype o phonon b anches and he es o hem a e plo in hal - anspa en colou o he sake o
cla i y. The non-symmo phic na u e (a glide plane in he basal plane) o ou space g oup is esponsible o he
wo old degene acy o modes a he X poin , whe e always wo phonon b anches a e s uck oge he and conse-
quen ly hei slope is no pe pendicula o he B illiouin zone bounda y, see X1 and X2 phonon modes in Fig.2.
The glide plane is also esponsible o he mixed cha ac e o longi udinal and ans e sal
1
and
2
b anches.
Only
3
and
4
a e pu e ans e sal wi h displacemen in he x-di ec ion. The
1
and
2
phonon b anches wi h
a mixed cha ac e o en end o c oss each o he , which leads o equen an ic ossings. The dispe sion cu es
o CePd2Al2 and LaPd2Al2 a e e y simila . Mos o hem appea a highe ene gies o CePd2Al2 compa ed o
LaPd2Al2. This can be unde s ood i we conside ha he CePd2Al2 uni cell has a smalle olume. The de ailed
discussion abou di e ences in hese wo homologs oge he wi h pa ial densi y o s a es is p esen ed in12.
The aim o ou expe imen al s udy is a mapping o he selec ed phonon dispe sion cu es and especially
modes wi h he A
1g
symme y as his symme y was assumed o be in ol ed in he magne o-elas ic coupling5.
The A
1g
phonon modes a e p esen only a he
Ŵ
poin , whe e we ind h ee o hem. Figu e3 shows he displace-
men ampli ude o a oms in he uni cell o all h ee A
1g
modes. The ene gy o he phonon mode, in ol ed in
he magne o-elas ic coupling, should be compa able wi h he ene gy o magne ic exci a ion om g ound s a e
in he unpe u bed le el scheme based on s anda d CEF heo y. Based on he magne ic exci a ions obse ed by
inelas ic neu on sca e ing5, his es ic s ou a en ion o he A
1g
mode a 12.8meV wi h a dominan mo ion
o Ce ions, see Fig.3. Di ec measu emen o inelas ic x- ay sca e ing a he
Ŵ
poin is no possible, because
o he s ong elas ic signal, he e o e he measu emen s we e pe o med in he close icini y o he
Ŵ
poin a
q=(0.00, −0.05, 0.00)
. The 0 1 8 B illiouin zone was selec ed o measu emen s as he calcula ed in ensi y o
A
1g
mode a 12.8meV is dominan in he spec a in his zone. Figu e4a shows he empe a u e dependence o
he
1
phonon mode (#13) ela ed by compa ibili y ela ions wi h he A
1g
mode a 12.8meV a he
Ŵ
poin . The
(3)

5A1
g+
3A2
u+
2B2
u×
(E
u+
A2
u
)
=
5E
u+
5E
g+
5A2
u+
3A1
g+
2B1
g.
Figu e1. Fi s B illiouin zones and labels o highe symme y poin s and di ec ions, wi h espec o18.
(a)Simple e agonal la ice (CePd2Al2). (b)Body cen e ed e agonal la ice (CeCuAl3). (c) Recip ocal body
cen e ed cubic la ice (CeAl2-di ec ace cen e ed cubic).
4
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esul s a e compa ed wi h calcula ed spec a in 4b. The measu ed spec a a he 0 1 8 B illouin zone a e almos
iden ical o bo h CePd2Al2 and LaPd2Al2. The empe a u e dependence o he phonon mode ene gy is negligible
wi h only a sligh shi o highe ene gy a low empe a u e. The spec al wid h o modes is simila a oom and
lowes empe a u e and also o CePd2Al2 and LaPd2Al2 homologs.
The o he phonon modes need also a close inspec ions, since he magne o-elas ic coupling in CeAl2, which
is he a che ypal example, a ec s no only he gi en phonon mode, bu he whole co esponding b anches. F om
his poin o iew, i is impo an o s udy he
q
dependencies o phonon b anches and compa e hem wi h he
calcula ed ones and wi h non-magne ic analogues. I is almos impossible o ind such a B illiouin zone, whe e
only he in ensi y o one phonon b anch will be in ense and well sepa a ed om he o he s. This p oblem is pa ly
sol ed by measu ing a se e al B illiouin zones a he same
q
posi ion. The ene gy o he phonon mode is hen
i ed and kep he same in all B illouin zones. The spec al line shape is gi en by a pseudo-Voig unc ion, which
pa ame e s we e de e mined by a i o he elas ic line. This p ocedu e imp o es a lo he da a analysis, bu s ill
he in ensi y a io be ween modes based on a dynamical s uc u e ac o has o be ixed o inc ease he s abili y o
he i . The example o such p ocedu e is shown in Fig.5 o CePd2Al2 compound,
q≈(0.00, 0.24, 0.00)
a 1.5K.
I has o be no ed ha he measu ed in ensi y is in eg a ed in ensi y om a pa o ecip ocal space and no an
in ensi y a gi en poin in ecip ocal space due o he ini e esolu ion. This could ha e signi ican in luence on
he measu ed in ensi y, especially whe e he in ensi y o he mode changes apidly wi hin he B illiouin zone.
This could also be he eason o enhanced in ensi y a ound 8meV in he spec a, seeFig.5. The same ea u e
was obse ed o bo h Ce and La homologs (see compa ison o measu ed da a in Supplemen a y ma e ial) and
canno be he e o e ela ed o he magne o-elas ic coupling.
Figu e2. Symme y analysis o 30 calcula ed phonon modes o (Ce,La)Pd2Al2, conside ing degene acy. The
abb e ia ion TL signi ies he mixed cha ac e o longi udinal and ans e sal pola isa ion and T signi ies a pu e
ans e sal cha ac e . Each panel highligh modes and b anches wi h selec i e symme y, he co esponding
labels a e abo e each panel. Fo de ails see chap e Resul s–Phonon dispe sion cu es in (Ce,La)Pd2Al2.
5
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The measu emen s we e pe o med a h ee di e en empe a u es 1.5, 7, and 280K o CePd2Al2 and 1.5,
80 and 300K o LaPd2Al2. The empe a u es 7 and 80K we e selec ed o be close o he s uc u al phase ansi-
ion, bu s ill abo e hem. The esul s o he abo e men ioned analysis a e shown in Figs.6 and 7 o bo h Ce
and La homologs, espec i ely. The q-dependences o phonon modes ag ee e y well wi h he calcula ed ones.
The empe a u e dependence o phonon ene gies is negligible, excep he
1
and
1
b anches a ound 15.7meV,
whe e he mo ion o Pd a oms is dominan . The signi ican di e ence be ween calcula ed and measu ed da a
is a ound 12.4meV
4
o LaPd2Al2, in con a y he esul s ag ee e y well o CePd2Al2 in his symme y. The
aw measu ed da a o Ce and La homologs a e almos iden ical (see Supplemen a y ma e ial), which sugges
be e ag eemen be ween measu emen and calcula ion o CePd2Al2. A signi ican shi o he
2
b anch o
he highe ene gy o a ound 14.0meV was obse ed o bo h compounds, al hough be e ag eemen is ound
again o CePd2Al2.
Figu e3. The a omic displacemen o a oms o he
Ŵ+
1
(A
1g
) phonon mode symme y. Calcula ed a 12 o
CePd2Al2. Fo cla i y he displacemen o each a om is 4 imes enla ged. The labels #13 #18 #30 co espond o
he no a ion used in Supplemen a y in o ma ion, which labels he modes o a gi en
q
wi h inc easing ene gy.
The ed, blue and g een colou s co espond o Ce, Pd and Al ions, espec i ely. The open ci cles ep esen s s a ic
a oms in he mode.
Figu e4. (a) The measu ed inelas ic spec a a he 0 1 8 B illouin zone (
q
=
(0.00,
−
0.05, 0.00)
) empe a u es
1.5 and 300K. The esul s a e shown o CePd2Al2 and LaPd2Al2 compounds. (b) The calcula ed in ensi y ( ull
black line) is he sum o all 30 modes a he 0 1 8 B illouin zone o CePd2Al2, he mos in ense modes a e
labelled.

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Discussion – quasi-bound s a e
The phonon dispe sion s udy in (Ce,La)Pd2Al2 is mo i a ed by he p oposed o ma ion o a quasi-bound s a e in
his compound. The e o e we discuss he expe imen al esul s wi h espec o he heo e ical model desc ibing
his quan um s a e.
The CEF magne ic exci a ions a e usually non-dispe si e wi hin B illouin zone, unlike phonons. The s eng h
o any magne o elas ic coupling,
gµ
0
, depends on
k
and on a gi en mode s. The mos impo an con ibu ion o
he in e ac ion e m can be exp essed as in8:
(4)
ˆ
H
in =−

lτ

k
sµ
g
µ
0(ks,τ)

ˆaks+ˆa
+
−ks
ˆ
Oµ(Rlτ)e
−ik·R
lτ
,
Figu e5. The i o measu ed da a a
q=(0.00, 0.24, 0.00)
on he le and calcula ed in ensi y based on
dynamical s uc u e ac o on he igh . Each pai o g aphs co espond o he gi en B illouin zone speci ied in
bold abo e hem. The mos in ense modes a e labelled. The colou o he selec ed mode is he same ac oss he
B illouin zones. The legend abo e shows modes o which ene gies we e possible o de e mine conclusi ely.
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whe e
l,τ
indexes un o e p imi i e uni cell and a oms in he basis, espec i ely. The
ˆa
and
ˆa+
a e he anni-
hila ion and c ea ion ope a o s o phonons. The ope a o
ˆ
Oµ
ep esen s CEF s a es and he
ˆ
Oµ
e
−ik·R
l
τ
should
ans o m as he phonon mode
ks
(as he IR o li le g oup). The
ˆ
Oµ
i sel hen ans o ms like IRs o a ac o
g oup o he li le g oup. The ac o g oup is a poin g oup in case o SSG, while i is no gene ally isomo phic
o a poin g oup in case o NSG, see chap e Resul s- Phonon dispe sion cu es in (Ce,La)Pd2Al2. This has a
consequence ha he di ec inding o
ˆ
Oµ
ope a o s o some high symme y poin s in he B illouin zone is no
possible in NSG. The si ua ion is much easie a he
Ŵ
poin (
k=0
), whe e he li le g oup is a poin g oup o
SSG and also NSG. This simpli ica ion is used in he non-dispe si e model o he in e ac ion e m8:
whe e
ˆa+
µ
and
ˆaµ
a e c ea ion and annihila ion ope a o s o phonons
µ
,
gµ
is a magne oelas ic coupling pa am-
e e . This in e ac ion e m was used o heo e ical desc ip ion in CeAl28, CeCuAl36, CeAuAl37 and CePd2Al25.
De e mina ion o he magne o-elas ic coupling o a ious phonon modes and he decision abou he domi-
nan con ibu ion is a highly non- i ial ask. In he case o cubic CeAl2 ( he i s B illouin zone is shown in
Fig.1c his was assis ed by expe imen al s udies o phonon dispe sion cu es. So ening and b oadening o
he
Ŵ
′
25
phonon mode was obse ed he e, sugges ing his mode o be conside ed in he model10. Such a di ec
obse a ion o a quasi-bound s a e in he phonon dispe sions o CePd2Al2, CeCuAl3 and CeAuAl3 has no been
epo ed, ye .
Be o e we u n o ou esul s on CePd2Al2, le s b ie ly discuss he case o CeCuAl3 and CeAuAl3. The p e ious
single c ys al s udies o he phonon spec a in hese wo compounds we e no ocused on speci ic phonon modes
wi h symme y conside ed in he heo e ical model. The c ys al s uc u e o CeAuAl3 was de e mined o be he
BaNiSn3 s uc u al ype19 wi h I4 mm space g oup. The space g oup is symmo phic and he poin g oup is C
4
.
The in e ac ion e m o model Hamil onian uses he
ˆ
O2
2
=ˆ
J
2
x−ˆ
J
2
y
ope a o 7. This means ha he co esponding
phonon modes should ans o m as
x2
−y2
basis unc ion, which co esponds o he
Ŵ3
(B1) phonon mode. Away
om he
Ŵ
poin , using he dispe si e model, such symme y could be ound in V, M poin s and

di ec ion,
because he co esponding ac o g oups a e isomo phic o he C
4
poin g oup, see Fig.1b. The c ys al s uc u e
o CeCuAl3 is o en epo ed he same as ha o CeAuAl3, so he conclusions based on symme y analysis would
be alid also he e. Howe e he c ys al s uc u e o CeCuAl3 seems o be mo e complex han CeAuAl3. The pa ial
si e mixing on Cu and Al Wycko posi ion was p e iously obse ed19–21. The model o a quasi-bound s a e is
based on he ac , ha Ce ions in he c ys al s uc u e ha e iden ical en i onmen s, which is no expec ed o he
(5)
ˆ
H
in
=−
µ
gµ
ˆ
aµ
+ˆ
a
+
µ
ˆ
Oµ
,
Figu e6. The compa ison o heo e ical calcula ion wi h he i o measu ed inelas ic X- ay sca e ing spec a
o CePd2Al2 in ene gy ange 0 - 17.5meV. The

, ○,
▽
symbols co espond o 1.5, 7.5, 280K, espec i ely. Fo
de ails o he i , see chap e Resul s – Phonon dispe sion cu es in (Ce,La)Pd2Al2.
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solid solu ion compounds. Also he s udy o magne ic exci a ions in he CeCu
x
Al
4−x
sys em by inelas ic neu on
sca e ing22 shows he con inuous e olu ion o mo e han h ee magne ic exci a ions in he spec a, which is he
sign o non-iden ical Ce en i onmen s.
In he case o CePd2Al2 he ele an phonon modes we e sugges ed o be he A
1g
modes5. We no e ha he
A
1g
IR ans o ms as
x2+y2
,
z2
basis unc ions in he s udied symme y D
4h
. Only sphe ical ha monics Y
0,0
and Y
2,0
o
l<9
ans o m as A
1g
, he e o e
ˆ
Oµ
could be conside ed as
ˆ
O0
2=
3
ˆ
J
2
z−
J(J
+
1
)
o CePd2Al2. The
si ua ion o ecip ocal space poin s away om he
Ŵ
poin is mo e complica ed, han in CeAuAl3, because he
P4/nmm is NSG. The ac o g oup is isomo phic wi h he poin g oups only a
Ŵ
, Z, V, W poin s and in he

di ec ion. The A
1g
ep esen a ion is p esen only a Z (poin g oup D
4h
).
Based on hese a gumen s, ou expe imen al s udy was ocused on he A
1g
modes and on modes in he

and

di ec ions. The e a e h ee A
1g
modes a
Ŵ
poin a 12.8, 15.7 and 36.6meV. Taking in o accoun ha he
ene gy o he mode should be compa able o he ene gy o CEF exci a ion om he g ound s a e, he mode a
36.6meV can be excluded. The measu emen o ela ed phonon b anches o A
1g
modes a 12.8, 15.7meV in he
icini y o he
Ŵ
poin doesn’ show any b oadening o so ening. The o he mode symme ies a he
Ŵ
poin (E
g
,
E
u
, A
2u
, B
1g
modes) in he ene gy in e al 7.5–17meV show a simila empe a u e dependence as he A
1g
modes,
o hei ene gy is cons an in empe a u e. Looking a he phonon dispe sion cu es in he

and

di ec ions,
we didn’ obse ed any so ening, which also says ha in he X and Z poin s he empe a u e beha iou is simila .
Compa ison o aw measu ed spec a in Supplemen a y ma e ials o CePd2Al2 and LaPd2Al2 homologues show
no signi ican di e ence, which could be asc ibed o he expec ed quasi-bound s a e.
The c ys al la ice o CePd2Al2 and LaPd2Al2 is dis o ed below 6K and 60K, espec i ely. A 1.5K, he la ge
dis o ion is ound o LaPd2Al2. The e o e we will discuss he in luence o dis o ion on phonon dispe sion
cu es in LaPd2Al2 homologue. Du ing he dis o ion we los he ou - old axis, jus wo- old axis emains. This
lowe ing o symme y leads o dec ease o dimension o IRs in mechanical ep esen a ion. I leads o he spli ing
o wo imes degene a ed phonon modes a he
Ŵ
poin (
Ŵ−
5
(E
u
),
Ŵ+
5
(E
g
)) and also in he

di ec ion (
5
). On
he o he hand in he

di ec ion he dis o ion leads only o he modi ica ion o hei
q
dependencies. In he X
poin he wo old-degene acy o phonon modes is kep , because i equi es he NSG o igin as in he e agonal
case. Fo una ely he e ec o he dis o ion is small and he di e ence is e y o en smalle han in compa ison
o Ce and La homologues (see Supplemen a y in o ma ion). The e o e we compa e he measu emen a low
empe a u e wi h phonon dispe sion cu es a oom empe a u e in he Figs.6 and 7 .
We can conclude ha ou measu emen s show e y good ag eemen o he obse ed phonon dispe sion
cu es wi h he calcula ed ones wi hou in oducing he quasi-bound s a e. The p e ious idea5 ha he A
1g
Figu e7. The compa ison o heo e ical calcula ion wi h he i o measu ed inelas ic X- ay sca e ing spec a
o LaPd2Al2 in ene gy ange 0 - 17.5meV. The

, ○,
▽
symbols co espond o 1.5, 80, 300K, espec i ely. Fo
de ails o he i , see chap e Resul s - Phonon dispe sion cu es in (Ce,La)Pd2Al2.
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phonon modes may be he s onges coupled o he magne ic exci a ions can be excluded by ou s udy. O he
mode symme ies a he
Ŵ
poin also seem o be una ec ed by he quasi-bound s a e, which opens he ques ion
o he meaning o applica ion o a dispe sion-less model and e en abou he p esence o a quasi-bound s a e in
CePd2Al2. Beside CePd2Al2 we explo ed he mode symme ies and iden i ied he high symme y poin s in he
i s B illouin zone o CeAuAl3 and CeCuAl3, a which he ollowing phonon s udy should be ocused in o de
o ollow he consequence o a non-dispe si e model o he possible quasi bound s a e.
Da a a ailibili y
Raw da a we e gene a ed a he SP ing-8 la ge-scale acili y. De i ed da a suppo ing he indings o his s udy a e
a ailable wi hin he a icle and i s Supplemen a y in o ma ion o : La ice dynamics in CePd2Al2 and LaPd2Al2.
Recei ed: 3 Augus 2021; Accep ed: 28 Sep embe 2021
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Acknowledgemen s
This wo k was suppo ed by he ollowing p ojec s: Czech Science Founda ion unde G an No. 17-04925J, ERDF
in he IT4Inno a ions&Nano echnology Cen e , CEET, VSB na ional supe compu ing cen e - pa h o exascale
p ojec (CZ.02.1.01/0.0/0.0/16_013/0001791) wi hin he OPRDE and he p ojec e-INFRA CZ (ID:90140) by he
Minis y o Educa ion, You h and Spo s o he Czech Republic, by he p ojec NanoCen -Nanoma e ials cen e
o ad anced applica ions, p ojec No. CZ.02.1.01/0.0/0.0/15_003/0000485, inanced by he ERDF and p ojec s
JSPS KAKENHI (G an Nos. JP16K05031, JP19H04408, JP20H01864). The inelas ic X- ay sca e ingexpe imen s
we e pe o med unde he app o al o JASRI (p oposal No. 2018A1676, 2018B1705) and inancially suppo ed by
he “Budding Resea che s Suppo ” p og am o JASRI.Sample p epa a ion and cha ac e isa ion was pe o med in
MGML (h p://mgml.eu/), which was suppo ed wi hin he p og am o Czech Resea ch In as uc u es (p ojec
no. LM2018096). We hank Adam Ba oš, Pe Če mák and Milan Klicpe a o aluable discussion and Ross
Ha ey Colman o p oo eading he a icle.
Au ho con ibu ions
P.J. supe ised he esea ch. P.D., K.C. and P.J. w o e he manusc ip . P.D. analysed he expe imen al da a and
pe o med he symme y analysis o phonon modes. P.C., S.T., K.K. and P.D. pe o med he expe imen al