Table Of ContentExtended Configurational Polyhedra Based onGraphRepresentation forCrystallineSolids
Koretaka Yuge1
1 DepartmentofMaterialsScienceandEngineering, KyotoUniversity, Sakyo, Kyoto606-8501, Japan
WeproposetheoreticalapproachbasedoncombinationofgraphtheoryandgeneralizedIsingmodel(GIM),
which enables systematic determination of extremal structures for crystalline solids without any information
about interactions or constituent elements. The conventional approach to find such set of structure typically
employsconfigurationalpolyhedra(CP)onconfigurationspacebasedonGIMdescription,whoseverticescan
always becandidates toexhibit maximum or minimum physical quantities. Wedemonstrate that the present
7
approachcanconstructextendedCPwhoseverticesnotonlyincludethosefoundinconventionalCP,butalso
1
includeothertopologicallyand/orconfigurationallycharacteristicstructuresonthesamedimensionalconfigu-
0
rationspacewiththesamesetoffigurescomposedofunderlyinglatticepoints,whichthereforehassignificant
2
advantageovertheconventionalapproach.
n
a
J I. INTRODUCTION mationaboutGIMpaircorrelations,butalsoincludeshigher-
2 dimensionalfigures(orlinks)consistingofthecorresponding
1 pairs. WewilldemonstratethattheproposedCPretainschar-
In classical systems where physical quantity (especially,
acteristicverticesfoundforconventionalCP,andalsocanfig-
] dynamicalvariables)canbealinearmapforstructures,aset
n ureoutotherspecialstructuresintermsofgraph,onthesame
of special structure that has maximumor minimumphysical
n low-dimensionalspaceconsidered.
quantity should always be restricted by spatial constraint on
-
s constituentsofgivensystem.Thiscanbequantitativelydeter-
i
d mined by the landscape of so-called configurationalpolyhe-
II. METHODOLOGY
. dra (CP),1 which determines maximum and minimum value
t
a that basis functions for describing structures (typically, gen-
m eralized Ising model2 (GIM) descrpition is employed) can Veryrecently,wesuccessfullyconstructunifiedrepresenta-
tion of microscopic structure on periodic lattice (i.e., atomic
- take: Suchspecialstructuresalwayslocateonverticesofthe
d configuration) in terms of both GIM and graph description.
CP.SincethelandscapeoftheCPcanbedeterminedwithout
n Explicit relationship between GIM and graph representation
anyinformationaboutenergyorconstituents,wecanapriori
o forgivenfigureRongivenlatticeisexpressedby4
c knowasetofcandidatestructuretoexhibitextremalphysical
[ quantitieswhenconditionofspatialconstraintonconstituents
NR
v1 odfictthioensyosfteamlloisyggivroeunn.dU-ssitnatgethstirsucchtuarreasctheraivseticasm,epflfiycbieenetnpirne-- hy (~s )iR=r R−1(cid:229)m Cm·Tr l(cid:229)∈RBl(~s )±tBl(~s )! ,(1)
7 vestigatedsofar,andtheconceptoftheCPhasrecentlybeen
5 extended to finite-temperature properties,3 where configura- where left-hand side correspondsto multisite correlation for
2 tionaldensityofstates(CDOS)alongCP-basedspecialcoor- figureR inGIMdescripition. MatrixB denotesuppertrian-
3 dinationwellcharacterizetemperaturedependenceofinternal gularpartofextendedgraphLaplacian,definedby
0
energynearorder-disorderphasetransition.
.
01 The practical problem for using CP is that number of B(a )(i,j)= f p(s i)f q(s j) (p,q∈a ,i,j∈R,i< j),(2)
7 its vertices exponentiallyincreases at high-dimensionalcon- R (0p (otherwise)
figuration space considered. It is thus fundamentally im-
1
: portant to find out as many extremal structures as possi- where p and q denotes conventionalGIM basis function in-
v ble at low-dimensional space based on geometrically low- dex on each lattice point i and j, and s is the pseudo-spin
i k
X dimensional figures, such as symmetry-nonequivalent pairs variable to specify atomic occupation. A=B+tB corre-
on lattice. However, it has been shown1 that a set of struc- sponds to adjacency matrix. N is the dimension of figure,
r R
a tureatverticesofCPisinvariantwithlineartransformationof C isintegerdependingonthetypeoffigure,andr denotes
m R
thecoordinationwithingivensubspace,whichdirectlymeans numberofthepossiblepathtoconstructconsideredfigureR.
thatwhenwewouldliketofurtherfigureoutextremalstruc- Here, we construct graph Laplacian (and adjacency matrix)
turesbased on the conventionalCP approach,we should ex- composedofsymmetry-equivalentneighboringedges:There-
plicitlyincludeinformationaboutlonger-rangeand/orhigher- fore,summationintheright-handsideistakenoverpossible
dimensional figures on lattice, but such CP projected onto pair figure l thatcorrespondsto subfigureof figure R. From
desired low-dimensional space typically lose significant in- Eq. (1), we have shown that information about higher-order
formation about vertices that are originally found at low- links(ormultisitecorrelationsforhigher-dimensionalfigure)
dimensional space based on low-dimensionalfigures. In or- inthestructurecanbeexplicitlyincludedinthelandscapeof
dertoovercomethisprobleminGIM-basedconventionalCP, graph spectrum composed of the correspondingpair figures.
thepresentstudyproposesconstructionofCPbasedonexten- ItisthusnaturallyexpectedthatwhenweconstructCPbased
dendgraphrepresentationunifiedwithGIM-baseddescription ontheextendedgraphrepresentation,wewouldobtainaddi-
forcrystallinesolids,whosespectrumnotonlycontainsinfor- tionalatomic configurationson vertices of the CP compared
2
with GIM-based CP. From Eq. (2), it is clear that adjacency (a) 4 L10 (b) 4 Z2
matrixforanygivenfigureRistraceless. Therefore,inorder
tfshoooarlscuAhbteaesre,oancfwteaeexlrltihezeeneirsgetiheveneemvllyaaplnilunodevyssecgsaftorpiargepagohtrefaedtpnhtheeorsgrcpeyhesautcrhtlartacauttnmeitsr.igztrhGeaeprsahuspucshmhpeaeocsnfterrrauegbgmy-- μ for A22- 202 "40" Z2 h energy for A- A1 2 - 202 L10T2 T3 2-(110)
uanladrictyorbreyspdoetnedrimnginainpgpluicpapteiornanhda/sobreleonwdeornbeoufonrdmooflietscuvlaelsuteo, -4 2B-(l1o1c0k) Grap -4 T1 "40" Block
relatetoitsenergetics.5–8
-4 -2 0 2 4 -4 -2 0 2 4
In the present study, we prepare examples of all possible μ2 for A1 Graph energy for A1
atomic configurations for A-B binary system on 4×4×4
expansion of fcc conventional unit cell having minimal unit (c) (d)
consistingofup16atoms,andcalculatecorrespondinggraph 4 "40" 4
csfoporencsoticrducemurep,daw.tiShoenprienofcvoAamri(apBbol)seietailtoemenasecxnht(.lAatFxtBiocr1e−tphxo)eisoneft,0ast.o5i,matnaickdec0so.7+n51figa(0ure)- or A3 02 gy for A- A1 3 02 "40" 2-(110)Z2
3rartdionnesa,rweset-cnaelicguhlbatoers(e3cNoNnd)-opradirer(im.eo.,mGeInMtmp2aoirfcAorfroerlautipontos μ f2-2 L10 Z2 aph ener-2 T1
upto3NNpair)toconstructconventionalCP,andalsocalcu- Gr-4 T3
lategraphenergyforlinearcombinationofthesamesetofA -4 L10
above. -4 -2 0 2 4 -4 -2 0 2 4
μ2 for A1 Graph energy for A1
FIG. 1: Configurational polyhedra (CP) at composition x =0.5.
III. RESULTSANDDISCUSSIONS (Left)ConvensionalCPintermsofsecond-ordermomentofA1,A2
andA ,corresponding toGIMcorrelationfor1-3NNpair. (Right)
3
ExtendedCPbasedontheproposedgraphspectrum,usinggraphen-
Figure 1 shows the constructed CP for convensionalGIM
ergyforA ,A andA .Structureattheverticesareemphasizedby
(left)andextendedgraph(right)representationatx=0.5,in 1 2 3
closedcircles.
termsofadjacencymatrix of1-3NN pairs. From Fig. 1(a),
we can find five ordered structures of L1 , ”40”, Z2, Block
0
and 2-(110) at the vertices. L1 and ”40” are known to be
0
ground-state atomic configurations for real alloys of Cu-Au
andPt-Rhsystem,9,10andZ2ofalternatingtwo-layerstacking
along(001)hasbeenpredictedasground-stateforPt-Rualloy
basedonsystematicfirst-principlesstudy11):Wecancertainly
confirmthatstructuresatverticesofCPcanbecandidatefor
ground-state structure. When we construct the extended CP
using linear combination of the same As, the resultant CP
is foundto have moreverticesthan the convensionalone, as
shownin Fig. 1 (b). We can clearlysee thatthreeadditional
structures of T1, T2 and T3 are found at the extended CP,
FIG. 2: Atomic configuration of nine ordered structures that are
and five structures at vertices of convensionalCP are all lo-
foundatverticesofconvensionaland/orextendedCPinFig.1.
catesatthose ofthe extendedCP, whichhasbeenconfirmed
in ourpreviousstudy. When we focusthe lower two figures
ofFig. 1 forCP of 2and 3NNpair figures, similar tendency
can be found: Three orderedstructuresof L10, ”40” and Z2 FromFig.3(a)and(c),wefindL12 andD022 orderedstruc-
can be found at vertices of both convensional and extended turesthathavebeenconsideredgroundstates forrealalloys,
CP,andtwoadditionalstructuresofT1andT3arealsofound and additional three ordered structures of Rhombo, Z1 and
atthoseoftheextendedCP.Figure2showsatomicconfigura- Block-2are found. Usingthe sameAs, theextendedCPs in
tionofthenineorderedstructuresatverticesofCPsinFig.1. Fig.3(b)and(d)successfullyretainsthesefiveorderedstruc-
These results certainly indicates that the proposed extended tures at their vertices, and also have additional four ordered
CPbasedongraphtheorycannotonlycaptureasetofstruc- structuresofU1,U2,U3andU4. Theseatomicarrangements
ture found at vertices of conventional CP, but also can find are illustrated in Fig. 4. Therefore, we can see that the pro-
other charcteristic structures in terms of graph (i.e., having posedextendedgraphrepresentationgenerallyretainsvertices
characteristiclinks). ofconventionalCPaswellasprovidesothercharacteristicor-
In order to see the composition dependence of these ten- deredstructuresusingthesamesetofpairfiguresonthesame
dencies, we alsoconstructconventionalandextendedCPsat dimensionofconfigurationspace,whichisadesiredproperty
x=0.75, based on A , A and A . Figure 3 shows the re- asdescribedabove.
1 2 3
sultant CPs, in terms of adjacency matrix of 1-3 NN pairs. The reason why the proposed representation can find out
3
(a) (b)
L12
4 4
Z1
A2 Block-2
or A2 02 D022 Z1 gy for A- 1 02
fµ2-2 Rhombo Block-2 aph ener-2 L12 U2 Rhombo
Gr
-4 -4
-4 -2 0 2 4 -4 -2 0 2 4
µ2 for A1 Graph energy for A1
FIG. 4: Atomic configuration of nine ordered structures that are
(c) (d) foundatverticesofconvensionaland/orextendedCPinFig.3.
4
Rhombo 4
A3
for Aµ23 02 D022 Block-2 energy for A- 1 - 202 D0R22homboU4 UZ11U3Block-2
-2 h
p
L12 Z1 Gra-4
-4
-6 L12
-4 -2 0 2 4 -6 -4 -2 0 2 4 morecharacteristicstructuresthanconventionalonecertainly
µ2 for A1 Graph energy for A1
reflectsthatthelandscapeoftheproposedgraphspectrumhas
more structural information (especially, higher-order struc-
FIG. 3: Configurational polyhedra (CP) at composition x=0.75.
(Left)ConvensionalCPintermsofsecond-ordermomentofA ,A turallinks)thatcannotbeexplicitlyincludedintheGIMde-
1 2
andA ,corresponding toGIMcorrelationfor1-3NNpair. (Right) scription. For instance, graph energy naturally include the
3
Extended CP based on the proposed graph spectrum, using graph information about asymmetric landscape of graph spectrum.
energyforA1,A2andA3.Structureattheverticesareemphasized When we quantitatively determine this asymmetry as third-
byclosedcircles. ordermomentofthespectrum,m ,thiscanbeexplicitlygiven
3
by,forinstance,
m [Spec(A −A )]=N−1(cid:229) (cid:229) (cid:229) i A j j A k k A i
3 1 3 13 13 13
i j k
(cid:10) (cid:12) (cid:12) (cid:11)(cid:10) (cid:12) (cid:12) (cid:11)(cid:10) (cid:12) (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
=m [Spec(A )]−m [Spec(A )]+N−1 (cid:229) hi|A |jihj|A |kihk|A |ii− (cid:229) hi|A |jihj|A |kihk|A |ii ,
3 1 3 3 13 13 13 13 13 13
(i,j,k∈(133) i,j,k∈(113) )
(3)
where A =A +A and A =A −A . Summation in which successfully results in the desired property found for
13 1 3 13 1 3
thethirdtermofright-handequationistakenoversymmetry- theextendedCPbasedongraphrepresentationcomparedwith
equivalenttriplet composed of one 1NN and two 3NN pairs theconventionalone,asseeninFigs.1and3.
(i,j,k∈(133)) and of two 1NN and one3NN pairs (i,j,k∈
(113)).
Fromtheaboveequation,itisnowclearthattakeingalin- IV. CONCLUSIONS
earcombinationofAsresultsincharacterizingotherhigher-
order links since m [Spec(A −A )] 6= m [Spec(A )]− Based on the graph theory and generalized Ising model,
3 1 3 3 1
m [Spec(A )]. More explicitly, differences in asymmetry we proposetheoreticalapproachto constructconfigurational
3 3
corresponds to difference in number of closed triplet links polyhedra(CP) forcrystallinesolids. TheextendedCPhave
composed of A element (i.e., element of s =+1) with one verticesnotonlyincludethosefoundinconventionalCP,but
(two)1NNandtwo(one)3NNpairs. Graphenergynotonly alsoincludeothercharacteristicstructuresonthesamedimen-
containsinformationaboutthirdordermoment,butalsocon- sional configurationspace with the same set of figures com-
tains second-ordermoment(correspondingto GIM pair cor- posedofunderlyinglatticepoints,whichthereforehassignif-
relations) as well as higher-ordermoment (correspondingto icantadvantageovertheconventionalapproach. We confirm
higher-orderstructurallinks composed of s =+1 element), thatthisdesiredpropertycanbenaturallyduetothefactthat
4
the proposedrepresentationhas more structuralinformation, GrantfromHitachiMetals·MaterialsScienceFoundation,and
especiallyabouthigher-orderstructurallinksforselectedele- Advanced Low Carbon Technology Research and Develop-
ment,thanconventionalGIMdescription. mentProgramof the JapanScience and TechnologyAgency
(JST).
Acknowledgement
This work was supported by a Grant-in-Aid for Scientific
Research (16K06704) from the MEXT of Japan, Research
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