Analytic Determination of Variance for Configurational Density of States in Crystalline Solids

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by  Tetyuya Taikei

AnalyticDetermination ofVariance forConfiguratinal Density ofStates inCrystallineSolids
Tetsuya Taikei,1 Kazuhito Takeuchi,1 and Koretaka Yuge1
1 DepartmentofMaterialsScienceandEngineering, KyotoUniversity, Sakyo, Kyoto606-8501, Japan
Our recent study elucidate that information of density of states in configuration space (CDOS) for non-
interacting system, characterized by spatial constraint on the system, plays essential role to determine ther-
modynamically equilibrium properties for interactingsystems. Particularlyfor disordered states, variance of
CDOSalongallpossiblecoordinationsplayssignificantrole. Despitethisfact,evenforbinarysystemofcrys-
tallinesolids,analyticexpressionforvarianceofCDOSasafunctionofcompositionhasnotbeenclarifiedso
7
far. HerewesuccessfullyderivevarianceofCDOSasafunctionofcompositionforpaircorrelations, whose
1
validityisdemonstratedbycomparingtheresultswithuniformsamplingofCDOSonreallattices.Thepresent
0
resultcertainlyadvancesdeterminingspecialmicroscopicstatetocharacterizeHelmholtzfreeenergyinclassi-
2
calsystems,whosestructureisdifficulttodeterminebynumericalsimulation.

n
a
J I. INTRODUCTION structureforGPstateisclearlygiveninmathematicalexpres-

0 sion,wedonotgiveexplicitformoftheGPstateforpractical
3 When the classical system is in equilibrium state, expec- lattice: This is because structure of GP state can be deter-

tation value of physical quantity, including dynamical vari- mined by partial average of configurationaldensity of states
]
n ables, can be typically obtained through well-known canon- (CDOS) along composition, which has not been analytically
n ical avarage, Q (x,T) = Z 1(cid:229) q(d)exp b E(d) , where knownevenfor paircorrelation. Especially, the GPstate re-
- r − d r − quiresinformationaboutvarianceofCDOSalongchosenco-
s
i summationistakenoverallmicroscopicsta(cid:16)tesatcom(cid:17)position ordinationasafunctionofcomposition.
d x on phase space. For purely-thermodynamicvarlablessuch In the present study, in order to overcome such problem,
t. asfreeenergy,wecannottakeensembleaverageappliedtody- we here focus on A B binary system on periodic lattice,
a x 1 x
m namicalvariables,sinceithasamemberofentropy.Sicenthe and provide analytical e−xpression of variance of CDOS for
number of possible microscopic states considered on phase pair correlation as a function of composition, which has not
d- space astronomically increases when number of constituents beenderivedsofar.Wedemonstratethevalidityofthederived
n increases,itbecomespracticallydifficulttodirectlyestimate, expression to compare with numerical simulation for binary
o especiallyfreeenergy,bythedefinitionofpartitionfunction. systemsonrepresentativelatticesincludingfccandbcc.
c Therefore, in orderto overcomesuch problem, severaltheo-
[ retical approacheshave been amply developedto effectively

1 sample important microscopic states to estimate dynamical II. DERIVATIONANDAPPLICATION
v and/oropure-thermodynamicalvariables,includingMetropo-
5 lisalgorism,entropicsamplingandWang-Landausampling.
In the present study, to describe pair correlation for given
6 Although the existing theoretical approaches successfully
5 predict physical quantities in equilibrium state when multi- atomicconfiguration~s onlattice,weemploygeneralizedIsing
8 model (GIM), whose basis function can completely specify
bodyinteractionsofthesystemisoncegiven,theydonotfo-
0 any possible atomic arrangements. Here, s corresponds to
cus on whether special microscopic state(s) independent of i
. spin variable to specify occupation of lattice site i by A of
1 information about energy and temperature can exist, which
0 characterized such equilibrium properties: In other words, s i=+1andBofs i=−1. IntheGIM,correlationfunction
7 forpairfiguremisgivenby
a set of special microscopic state to determine equilibrium
1
: propertiesisunknownaprioriwithoutinformationaboutcon- x (~s )= s s , (1)
v stituentelements,multibodyinteractionsortemperature,lead- m h i kim,lattice
i
X ing to finding a optimal set of such states by existing ap-
where denotestakingarithmeticaverageoverallpair
proaches. h·im,lattice
r site i and k forms pair m. In order to consider variance of
a Very recently, we focus on the role of spatial constraint
CDOSalongpairm,werewriteEq.(1)inmoreexplicitform:
onconstituentsinclassicalsystem(e.g.,latticeforcrystalline
solids, and volumeand density for liquidsin rigid box), and x m(~s )=(2DN)−1(cid:229) gm(i,k)s i(~s )s k(~s ), (2)
successfully find a set of special microscopic states, whose
i,k
structuredependsonlyontheclassofspatialconstraintandis
independentofconstituents,interactionsandtemperature:In- whereN andDrespectivelydenotesnumberoflatticepoints
formation about physical quantity for the special states can andnumberofpairmpersite,g (i,k)=1ifsiteiandkforms
m
well characterize equilibrium properties including free en- pairmandg (i,k)=0forotherwise,andsummationistaken
m
ergy,internalenergyandelasticmodulus. Especiallyforesti- overalllatticepointsinthesystem.Notethatwehereconsider
matingfreeenergy,correspondingsinglespecialstate,called thatalllatticepointforemptylatticeissymmetry-equivalent,
GrandProjection(GP)state,shouldbefundamentallyimpor- whichholdsforrepresentativelatticesincludingfcc,bccand
tantsinceithasbeenconsideredthatinclassicalsystem,free diamond.Usingtheaboveequation,varianceofCDOSalong
energycannotbedescribedbyinformationaboutasinglemi- pairmcanbeexplicitlygivenby
croscopicstate. In ourpreviousstudy, althoughconditionof
2
hx mivar = x m2 ~s −hx mi~s2 =( 4D2N2 −1(cid:229)i,k(cid:229)p,qgm(i,k)gm(p,q) s is ks ps q ~s )−(2x−1)4, (3)
(cid:10) (cid:11) (cid:0) (cid:1) (cid:10) (cid:11)
where denotestakingaverageoverallpossibleatomicar- shouldbesatisfied,where x denotesvarianceofCDOS
h·i~s h mivar
rangementatcompositionxconsidered.Thelastequationcan alongpairm consideringwholecompositionsimultaneously.
be obtained since g (r,s) is independentof atomic arrange- It is clear that the above condition holds true only at
m
pmaeirntma.nSdinitcehaesxabceteenxsphroeswsnionthoaft hxx mi~s =is(e2xxp−ec1te)2dftoorhaanvye weqeuciaatnomgiivceceoxmplpicoistietixopnr,exss=ion1/fo2r. tUhesifinrgstthteersmeporferriegqhuti-shiatensd,
significantlycomplexfunctionofbhotmhiNvarandx,weherefocus sideofEq.(3):
oncapturinggloballandscapeof x forsufficientlylarge
Nac.oFnrtoimnuEouqs.(f3u)n,cwtieoncaonfsceoemtphoatsiwhtiohmneinxva,hrxitmiisvarreaissoenxapbrelyssgeidveans (cid:229)i,k(cid:229)p,qgm(i,k)gm(p,q) s is ks ps q ~s
bypolynomialofuptodegree4,since x m2 ~s isgivenbyupto =2(cid:229) gm(i,k)2 s i2s k2(cid:10)~s (cid:11)
foursite correlation. Underthiscondition,wecanstartfrom i,k
expressingvarianceofCDOSalongpa(cid:10)irm(cid:11)as + (cid:229) g (i,k)(cid:10)g (k,q(cid:11)) s s 2s
m m i k q ~s
i,k=p,q
hx mivar(x)= f(x)=i(cid:229)=40aixi, (4) + (cid:229)i6=q gm(i,k)gm(p,q) (cid:10)s is ks ps(cid:11)q
i,k,p,q
wherecoefficientsascandependonN andD. Sincewehave i=p,q (cid:10) (cid:11)
i k6=p,q
fiveunknownquantitiesa , ,a ,weshouldrequirefivein- 6
dependentconditions for0x··· .4 Since at x=0 and x=1, =2DN 2+2DN 2(2D 1)(2x 1)2
we have respectively a sinhglmeiavatromic arrangement(i.e., pure +2DN · 2DN 2·(2D −1) 2 (−2x 1)4. (8)
·{ − − − } −
AandB),weget
Inthelastequation,factor2inthefirsttermcomesfrompos-
f(0)= f(1)=0. (5) sible permutations of i= p,k = q and i=q,k = p. When
we substitute Eq. (8) into Eq. (3), we can obtain varianceof
Since pair correlation, hs is kim,lattice, is symmetric by ex- CDOSatx=1/2as
changing all A and B constituents, x (x) is symmetric
twaiinthrespecttox=1/2. Applyingthhismciovnadrition,wecanob- hx mivar 21 =(DN)−1. (9)
(cid:18) (cid:19)
df Note that the formof Eq. (9) is the same as variance of pair
= 0
dx correlationforrandomalloyatequiatomiccompositionprevi-
(cid:12)x=1/2
(cid:12) ouslyshown,8 althoughtheprerequisitesandexplicitexpres-
d(cid:12)f df
(cid:12) = . (6) sion for four spin product, including Eqs. (7) and (8) in the
dx dx
(cid:12)x=0 (cid:12)x=1 previousstudyremainsunclear.
(cid:12) (cid:12)
From Eqs. (5) and (6), (cid:12)we have obt(cid:12)ained four independent UsingEqs.(5),(6)and(9),wecanfinallyobtainfiveinde-
(cid:12) (cid:12) pendentconditions,andthus,wecanprovideanalyticexpres-
conditions, and therefore, we should require one remaining
sionforvarianceofCDOSas
condition. Eqs. (5) and (6) do not contain any information
about N or D, which naturally indicates that the remaining 1
x (x)= 16x4 32x3+16x2 , (10)
condition should be obtained by explicitly consider Eq. (3). h mivar DN −
In the present study, we assume that at N ¥ , configura-
(cid:0) (cid:1)
tional average of four spin product, s s s →s , is treated which means that corresponding standard deviation is given
i k p q ~s bysimplefunctionof
bytakingproductoverindependentlyoccupiedspinvariables
(cid:10) (cid:11)
depending only on composition x. This would be allowed
4x(1 x)
when we consider atomic arrangements over whole compo- hx misd(x)= √D−N . (11)
sition, since underthiscondition,wecanindependentlygive
spinvariableof+1or-1ateachlatticesitetoconstructIRC.
Inordertodemonstratethevalidityofderivedstandardde-
However,wehereconsiderthathx mivar atfixed composition. viation, Eq. (11), we performMonte Carlo (MC) simulation
Therefore,inordertoapplyourassumptionforthefourspin uniformly sampling possible atomic arrangements for 64-,
product,thecondition 128-and256-atombinarysystemonfccandbcclatticesalong
1NNand2NNpairs,toobtainxandN dependenceof x
Nli→m¥ hx mivar(x)=Nli→m¥ hx mivar (7) forpracticallattices. Figure1 showsthe resultanthx mihsdmfiosdr
3
fcc bcc numericalsimulationexhibitnegativedeviationfromEq.(11)
0.08
0.08 N=64 2NN N=64 2NN anddeviationdecreaseswithincreaseofN, 2NNpaironbcc
exhibitsbothpositiveandnegativedeviationdependingonN.
0.06
0.06 1NN Finally,wenotethatthepresentresultsofanalyticexpres-
0.04 1NN 0.04 sion of hx misd for pairscertainly advancedeterminationof a
special microscopic state, which we call ”Grand Projection
0.02 0.02 state (GP state)”, whose physical quantity characterize free
energylandscapeoverwhole compositionin thermodynami-
0 0
Standard deviation for ppair 00000.....00000 012345 N=128 N=128 00000....00001234 Standard deviation for pair fxrcpcsoethaoa≤lrlositlepeiyst0seoio.oenso5nqnsnciu,fgbootiwihllvoreiehberbpdnirCicaiinunlhDiaarmastrOdttyiioioSscrvsteneaeyac,trtsitewsltmsyeueowmuffmsfiel.hctleciollpHai-aueslnsoenmlssdwticlcotokyehaovmnallteoathsprrweyt,gordsceesptioaenaNttnimermociawtnseeri.saphnelotTetqarfooouvexffiecCa≥trpohtaDoaneg0rmsOet.Gti5riSaocuPlafcancahstlodvoxtGam/enmotrPgei--r
N=256 ageattheequiatomiccompositionisnon-trivialfornumerical
0.03 0.03 simulation. Therefore, analytic determinationof CDOS (es-
pecially,upto2nd-ordermoment,correspondingto x we
0.02 0.02 consideredhere)asafunctionofcomposition,Eq.(1h1)m,iissdsig-
nificantlyusefultoconstructtheGPstatewithoutperforming
0.01 0.01
undesirablenumericalsimulation.
0 0
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
Composition, x
III. CONCLUSIONS
FIG. 1: Standard deviation of CDOS along nearest neighbor pair
on fcc (left) and bcc (right) lattice, as a function of composition x
Forcrystallinesolids,wederiveanalyticexpressionofvari-
andnumberofatoms,N. Opencircleandtrianglesdenoteresultsof
anceforconfigurationaldensityofstates(CDOS)alongcho-
numerical simulationfor1NNand2NN pair, andsolidandbroken
curves denote those of derived expression, Eq. (11), for 1NN and sen pair, as a function of composition. We demonstrate the
2NNpair. validityofthederivedexpressionbycomparingtheresultsfor
nearestandsecond-nearestneighborpairsonfccandbcclat-
tices,wherewefindsuccessfulagreementbetweentheexpres-
sionandsimulationwithincreaseofsystemsize. Thepresent
numerial simulation compared with the derived analytic ex- resultcanbe significantlyusefulto determinespecialmicro-
pression of Eq. (11). Here, D for fcc 1NN, 2NN, and bcc scopic state to characterize free energylandscape, which we
1NNand2NNpairrespectivelytakes6,3,4and3. recentlyfindforclassicalsystems,withoutperformingunde-
WecanclearlyseefromFig.1thatwhennumberofatoms sirednumericalsimulation.
in the system increases, standard deviation for CDOS along
chosen pair by numerical simulation unversally agree with
ourderivedexpression. ParticularlyatN=256,standardde-
Acknowledgement
viation given by Eq. (11) exhibits excellent agreement with
numerical simulation: We also confirm that when we fur-
ther increase number of atoms in the system, deviation be- This work was supported by a Grant-in-Aid for Scientific
tween Eq. (11) and numerical simulation is within accuracy Research (16K06704) from the MEXT of Japan, Research
of the simulation. Thus, validity of Eq. (11) is now well- GrantfromHitachiMetalsMaterialsScienceFoundation,and
·
demonstrated. Additional importantpoint is that asymptotic Advanced Low Carbon Technology Research and Develop-
behavior of standard deviation to Eq. (11) does not always mentProgramof the JapanScience and TechnologyAgency
monotonic:Forinstance,while x for1NNpaironfccin (JST).
h misd
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