Calculation of the soft-mode frequency for the alpha – beta transition in quartz

Optik127(2016)4470–4472

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Optik

j ou rn a l h o m e p a g e :w w w . e l s e v i e r . d e / i j l e o

Calculation

of

the

soft-mode

frequency

for

the

alpha

–

beta

transition

in

quartz

H.

Yurtseven

a,∗

_,

_O.

_Tari

b

a_{DepartmentofPhysics,MiddleEastTechnicalUniversity,06531Ankara,Turkey}

b_{DepartmentofMathematicsandComputerScience,IstanbulArelUniversity,Büyükc¸ekmece,34537Istanbul,Turkey}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received21December2015 Accepted19January2016 Keywords: Softmode Alpha-betatransition Quartz

a

b

s

t

r

a

c

t

The␣–␤structuraltransitionoccursinquartzatTC=846K.Thefrequencyofthesoftmode

associ-atedwiththevolumeincrease,decreaseswithincreasingtemperatureasthetransitiontemperatureis

approached.

Inthisstudy,wecalculatethesoft-modefrequencyasafunctionoftemperatureusingthevolumedata

bymeansofthemodeGrüneisenparameterforthe␣–␤transitioninquartz.

Ourcalculatedfrequenciesofthesoft-modeagreewiththeobserveddatafromtheliterature.This

showsthatthemethodofcalculatingthesoftmodefrequencyfromthecrystalvolumeisadequate,

whichcanexplainthesoftmodebehaviorassociatedwiththe␣–␤transitioninquartz.

©2016ElsevierGmbH.Allrightsreserved.

1. Introduction

Quartz exhibits a phase transition by lowering the sym-metry from the high-temperature hexagonal ␤ phase to the low-temperaturetrigonal _␣ phase at 847.5K [1]. It wasfound someyears agothat aweakmodeofsymmetry A1 with

room-temperatureRamanshiftof147cm−1growsinintensitybymoving towardzeroRamanshiftasthe␣–␤transitionisapproached[2]. Itwasstatedthatthissoftmodeplaysthefundamentalroleinthe ␣–␤transitioninquartz[2,3].Sincetheincommensurate(INC) phaseoccursinanarrowtemperaturerangeof∼1.3Kbetweenthe ␣and␤phases,itsoccurrencehasbeenattributedtothecoupling ofthesoftmodewithTAmodesinquartz[4].Asthesymmetry changesbetweenthephases,manylow-frequencyhigh-amplitude modesofvibrationareexcited,whichcausesorientational disor-deroftheSiO4tetrahedrainquartz[5].Thesoftmodeassociated

withthephasetransitionsinquartzpropagatesasaphononthat movestheSiO4tetrahedraasrigidunits[6].Thesoftmodebehavior

ofthe␣–␤transitioninquartzhasbeenstudiedextensively[7–9], whichisassociatedwiththeSiO4rotation–vibration[10]asstated

above.UsingthetemperaturedependenceoftheobservedRaman frequenciesforthesoftmode(147cm−1)andalsothe207cm−1 mode[2],wehavecalculatedtheRamanlinewidthsofthose lat-ticemodesclosetothe␣–␤transitioninquartz[11].Inanother study[12],wehavecalculatedthetemperaturedependenceofthe

∗ Correspondingauthor.Tel.:+903122105056. E-mailaddress:hamit@metu.edu.tr(H.Yurtseven).

Ramanfrequencyshiftsandthelinewidthsfortheopticallattice vibrations(128cm−1and466cm−1)inthe␣phaseofquartzusing theanharmonicselfenergymodel.

Temperature[10,13,14]andpressure[15,16]dependencesof thevolumehavealsobeencalculatedusingthemoleculardynamics (MD)forthe␣–␤transitioninquartz,asmeasuredexperimentally [17–21].

Vibrationalfrequenciescanbecalculatedfromthecrystal vol-umethroughthemodeGrüneisenparameter.Thiscalculationcan alsobeperformedforthe␣–␤transitionin quartz.Usingthe observedvolumedata(neutrondiffractionandtheunit-cell vol-umeoftheaveragestructure)[20],wehavecalculatedtheRaman frequencyofthe207cm−1modeatvarioustemperaturesforthe␣ –␤transitioninquartz[22]andalsofortheRamanfrequencyof aninternalmodeforSiO2-moganite[23]usingtheunitcellvolume

[24].Veryrecently,wehavecalculated[25]theresonantfrequency fromtheneutrondiffraction[26]andvariationofvolume[20]with thetemperatureinthevicinityofthe_␣–_␤transitioninquartz.

Inthisstudy,thesoftmodefrequency(147cm−1)iscalculatedas afunctionoftemperatureusingthevolumedata[7–9]throughthe modeGrüneisenparameterclosetothe␣–␤transitioninquartz. Below,inSection2wegiveourcalculationsandresults.In Sec-tion3,wediscussourresults.ConclusionsaregiveninSection4.

2. Calculationsandresults

We calculated the soft-mode frequency using the volume data through the mode Grüneisen parameter for the ␣ – ␤ http://dx.doi.org/10.1016/j.ijleo.2016.01.167

H.Yurtseven,O.Tari/Optik127(2016)4470–4472 4471

Fig.1. Observed soft-mode frequency[7–9] as a function ofvolume [17,19]

extractedfromEq.(1),asalsogivenpreviously[10]intermsofthevolumeincrement (%)forthe␣–␤transitioninquartz.

transition in quartz. The volume increment can be defined as [10],

ı= _VV−V␣

␤−V␣ (1)

where,V␣istheequilibrium␣-quartzwithcellvolumeandV␤is

theunitcellvolumeatthe␣–␤transitioninquartz.Usingthe observedvaluesofV␣(T=0)andV␤(T=TC)[19],thevolumevalues

wereextractedfromthevaluesofthevolumeincrement(%)[10] accordingtoEq.(1),asgiveninTable1.Astheobserved[7–9] soft-modefrequencyplottedasafunctionofthevolumeincrement(%) previously[10],weplotherethesoft-modefrequencyasafunction ofvolumeforthe␣–␤transitioninquartzinFig.1.Fromourplot, weobtainedthesoft-modefrequencywiththevolumeas

=a+bV+cV2 ₍₂₎

where,a,bandcareconstants,whicharegiveninTable1.Thus, using the observed values of V␣(T=0) and V␤(T=TC) as given

above,thecorrespondingvaluesofthesoft-modefrequencywere obtainedthroughEq.(2)as␣(T=0) and␤(T=TC) forthe␣–␤

transitioninquartz,asgiveninTable1.

ThemodeGrüneisenparameterofthesoftmodecanbe deter-minedfromthefrequencyshiftsandvolumechangebydefining =−V

 d

dV (3)

Accordingtothevariationofthefrequencywiththevolume (Eq.(2))usingtheobserveddataforthesoft-modefrequency[7–9] andforthevolumedataextractedfromEq.(1)(Fig.1),thevalue canbedeterminedforthesoftmodefrequencyinthe␣-phaseof quartz.Bydefiningthefrequencyalteration_ısimilartothevolume incrementıinEq.(3)as,

1−ı₌ −␣ ␤−␣ (4) or ı=



␤−



/



␤−␣



(5)

valuesofthemodeGrüneisenparametercanbedeterminedasa ratio,

= V_·ı

ı (6)

ThevaluesofthemodeGrüneisenparameterweredetermined atvarioustemperaturesfromthevolumeandfrequencydataatthe temperaturesofT=0andT=TCaccordingtoEq.(6).The

tempera-turedependenceof_isplottedinFig.2forthe_␣-phaseofquartz. OncewedeterminedthemodeGrüneisenparameterfromthe vari-ationofthesoft-modefrequencyandthevolumechangeaccording

T (K) 900 800 700 600 500 400 300 200 γγ 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6

Fig.2.VariationofthemodeGrüneisenparameterwiththetemperatureforthe ␣phaseofquartzaccordingtoEq.(6).

T (K) 800 700 600 500 400 300 200 νν (c m ) 0 50 100 150 200 250 Observed [7 - 9] Calculated (Eq. 7)

Fig.3. TheRamanfrequenciescalculated(Eq.(7))forthesoftmodeasafunctionof temperatureforthe␣–␤transitioninquartz.

toEq.(6),thesoft-modefrequencycanthenbecalculatedatvarious temperaturesusingthevolumedatathroughthemodeGrüneisen parameteraccordingtotherelation

=A (T) +0exp



−ln



V (T) /V0



(7) which can be obtained from Eq. (3) with the additional temperature-dependentA(T)term.Thistermcanbeassumedas

A (T) =a0+a1T (8)

where,a0anda1areconstants.Valuesofthosecoefficientswere

determinedbyfittingEq.(7)totheobservedfrequencies[7–9]as giveninTable1.

Finally, using the observed data [10], values of the mode Grüneisenparameterwasgenerated(Fig.2),V0=V_␤and0=_␤

withthevaluesofthecoefficientsa0anda1(Table1),thesoft-mode

frequencieswerepredictedasafunctionoftemperatureaccording toEq.(7)forthe␣–␤transitioninquartz,asplottedinFig.3. 3. Discussion

Thesoft-modefrequencywaspredictedasafunctionof temper-atureusingtheobservedvolumedata[10]forthe␣–␤transition inquartz, asplottedinFig.3.By fittingEq.(7) totheobserved dataforthesoft-modefrequency[7–9],thefittedparametersofa0

anda1forthetemperature-dependenttermA(T)weredetermined

(Table1).The temperaturedependence ofthe modeGrüneisen parameter()wasobtainedtopredictthesoftmodefrequencies, asplottedinFig.2.Itdecreasesasthetemperatureincreasesup tothetransitiontemperature(TC=846.5K)inquartz,asexpected.

4472 H.Yurtseven,O.Tari/Optik127(2016)4470–4472

Table1

Valuesofthecoefficientsa,bandc(Eq.(2)),andthecoefficientsofa0anda1(Eq.(8))forthevariationofthesoft-modefrequencywiththevolumeforthe␣–␤transition

inquartz(seeFig.1).ValuesofthevolumeandthesoftmodefrequencyatT=0andT=TCarealsogivenhere.

a(cm−1_{) b(cm}−1_/Å3_{) c(cm}−1_/Å6_{) V}

␣(Å3)(T=0) V␤(Å3)(T=TC) ␣(cm−1)(T=0) ␤(cm−1)(T=TC) a0(cm−1) a1(cm−1/K)

13,669.82 −196.02 0.68 110.73 118.03 269.45 34.58 292.95 −0.394

volume(Fig.1),thisdecreaseinthemodeGrüneisenparameter withthetemperature(Fig.2),leadstothesoftmodefrequency decreasingaccordingtoEq.(7)towardthetransitiontemperature TC,asalsoobservedexperimentally[7–10].Asseenfromthis

fig-ure,ourcalculatedRamanfrequenciesofthesoftmodegetlower incomparisonwiththeobserveddata[7–9]asthetransition tem-peratureisapproachedaboveabout600Kinthe␣phaseofquartz. So,theagreementwiththeexperimentaldataismuchbetteratlow temperatures.ThisdiscrepancynearTCmaybeduetothecritical

behaviorofthesoftmodefrequency,whichisalsoreflectedonthe anomalouschangeinvolume.However,thisanomalousbehavioris nottakenintoaccountinEq.(7)topredictthesoftmodefrequency fromthevolumedata.Apower-lawanalysisofthesoftmode fre-quencyandthevolumemightbeneedednearTCtodescribethe

anomalousbehaviorwiththecriticalexponent.

Calculationof themodeGrüneisen parameter forthe soft modeas a functionof temperaturewasperformedbasicallyby using theobserved values of the volume (V␣ and V␤) and the

soft mode frequency



␣and␤



at the minimum (T=0) and maximum(TC=846.5K)temperatures.Betweenthetwoextreme

temperatures,usingthevolumeandsoftmodefrequencydata,the temperaturedependenceofthemodeGrüneisenparameterwas generated.

Our  values which decrease between1.37and 0.14 asthe temperatureincreasesfromabout250Kupto800K,respectively, canbecomparedwithourvalueofP=0.75astheisobaricmode

GrüneisenparameteroftheRamaninternalmodeof501cm−1for the␣–␤transitioninSiO2-moganite[23].However,forthesoft

modeof207cm−1(atroomtemperature)weobtainedinourrecent study[22]thevaluesofP=5.5usingtheneutrondiffractiondata

forthevolume[20]andP=2.5usingtheunit-cellvolumedata

oftheaveragestructureofquartz[20].ThosePvaluesaremuch

higherthanthetemperature-dependent(T)valuesvaryingfrom 1.37to0.14asstatedabove,forthesoftmodeof207cm−1,which wealsostudiedhere.Theremightbetworeasonsforthis discrep-ancyregardingthevaluesofthemodeGrüneisenparameterofthe samesoftmode(207cm−1)forthe␣–␤transitioninquartz.Firstly, forourrecentstudy[22],weusedconstantvaluesofPtopredict

theRamanfrequenciesofthe207cm−1modeusingtheobserved volumedata[8]fromtwodifferentsources.Secondly,weusedthe observedfrequencydata[2] todeterminetheP values forthe

207cm−1modeinourpreviousstudy[22].

Inthisstudy,differently fromourearlierstudies[22,23],we calculatedtheRamanfrequencyofthesoftmode(207cm−1)by consideringthetemperaturedependenceofthemodeGrüneisen parameterforthissoftmode(Fig.2)usingthedifferentsources oftheobservedvolumedata[17,19]andthesoft-modefrequency data[7–10].Thereisalsocontradictionintheliteraturewhether theroom-temperaturerenormalizedsoftmodeisat147cm−1[2] orat208cm−1[3].Overall,themethodofpredictingthesoft-mode frequency using thevolume data throughthe modeGrüneisen

parameterasgiveninourearlierstudies[22,23]andalsointhis study,leadstopredicttheobservedbehaviorofthismode ade-quately for the␣ – ␤transition in quartz. Calculatingthe soft modefrequenciesbyconsideringthetemperaturedependenceof thesoft-modeGrüneisenparameteraswestudiedhere,cangive betterinsightintothemechanismofthe␣–␤transitioninquartz. 4. Conclusions

Thesoft-modefrequencywaspredictedfromthevolumedata byregardingthetemperaturedependenceofthemodeGrüneisen parameterforthe␣–␤transitioninquartz.Ourresultsshowthat thevaluesvaryfrom1.37to0.14asthetransitiontemperature (TC=846.5K)isapproachedfromthelow-temperature␣-phasein

quartz.Usingthevaluesdetermined,thesoft-modefrequencies werepredictedfromthevolumedatabymeansofthefitting pro-cedure.

TheRamanfrequenciesofthesoftmodewhichweobtained, indicatethattheorder–disordertransition(␣–␤transition)in quartzisassociatedwiththecriticalbehaviorofthesoftmode stud-iedhere.ThemethodofcalculatingtheRamanfrequenciesfrom thevolumedatathroughthemodeGrüneisenparameteratvarious temperaturesaspresentedhereforthe␣–␤transitioninquartz, canalsobeappliedtosomeothermolecularcrystalsexhibitingthe softmodebehavior.

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