Far-infrared to millimeter astrophysical dust emission I: A model based on physical properties of amorphous solids

0.37 MB·English

by  C. Meny

Astronomy&Astrophysicsmanuscriptno.meny-1 February5,2008
(DOI:willbeinsertedbyhandlater)
Far-infrared to millimeter astrophysical dust emission
I: A model based on physical properties of amorphous solids
7
0
0 C.Meny1,V.Gromov1,2,N.Boudet1,J.-Ph.Bernard1,D.Paradis1,andC.Nayral1
2

n 1 Centred’EtudeSpatialedesRayonnements,CNRS,9,AvenueduColonelRoche,31028Toulouse,France
a email:[email protected],[email protected]
J
2 SpaceResearchInstitute,RAS,84/32Profsoyuznaya,117810Moscow,Russia

9

1
ReceivedJuin2006/Acceptednovember2006?
v
6
2 Abstract.Anabstractshouldbegiven
2
1
0
We propose a new description of astronomicaldust emis- 1997, Draine & Li 2001). Most of them require a wide range
7
sion in the spectralregionfromtheFar-Infraredto millimeter ofthedustgrainsizesspanningalmost3ordersofmagnitude,
0
/ wavelengths. Unlike previous classical models, this descrip- fromsub-nanometerdimensionstofractionsofamicron.They
h
tion explicitly incorporates the effect of the disordered inter- also require a minimum of three components with different
p
nal structure of amorphous dust grains. The model is based chemical composition in order to explain the ultraviolet and
-
o onresultsfromsolidstatephysics,usedtointerpretlaboratory opticalextinction,alongwithinfraredemission.
r data.Themodeltakesintoaccounttheeffectofabsorptionby
t
s DisorderedChargeDistribution,aswellastheeffectofabsorp- The smallest component is needed to explain the ”aro-
a matic” emission features at 3.3, 6.2, 7.7, 8.6 and 11.3µm.
: tion by localized Two Level Systems. We review constraints
v It is now routinely assigned to large aromatic molecules or
on the various free parameters of the model from theory and
i Polycyclic Aromatic Hydrocarbons (PAH) (Leger & Puget
X laboratoryexperimentaldata.Weshowthat,forrealisticvalues
1984, Allamandola et al. 1985, De´sert et al. 1990). PAH are
r ofthefreeparameters,theshapeoftheemissionspectrumwill
a transientlyheated afterabsorbingsingle UV andfar-UV pho-
exhibitverybroadstructureswhichshapewillchangewiththe
tons,andcooldownemittingNIRphotonsinvibrationaltran-
temperature of dust grains in a non trivial way. The spectral
sitionsrepresentativeoftheiraromaticstructure.
shape also depends upon the parameters describing the inter-
nal structure of the grains. This opens new perspectives as to
A second component, composed of Very Small Grains
identifyingthe natureof astronomicaldustfrom the observed
(VSG)isrequiredtoexplainthecontinuumemissionintheMid
shape ofthe FIR/mm emission spectrum.A companionpaper
Infra-Red (MIR). Emission in this range still requires signifi-
will provide an explicit comparison of the model with astro-
canttemperaturefluctuationsofthegrains,whichnecessitates,
nomicaldata.
underconditionsprevailingintheInter-StellarMedium(ISM),
grainssizesinthenanometerrange.VSGscouldbecomposed
ofcarbonaceousmaterialwhoseabsorptioncouldalsoexplain
1. Introduction
the 2200ÅUV bump in the extinctioncurve(see for instance
ItisnowwellestablishedthattheSpectralEnergyDistribution De´sertetal.1990).
(SED)fromtheInter-StellarMedium(ISM)emissionfromour
A third component, composed of Big Grains (BG), with
Galaxy and most external galaxies is dominated by thermal
sizes from10-20nmto about100nm,is necessaryto account
emission from dust grains which spans over almost 3 orders
forthelongwavelengthemission(inparticularasobservedin
of magnitude in wavelengths, from the Near Infra-Red(NIR)
theIRAS100µmbandandabove).TheBGcomponentdomi-
to the millimeter wavelengthrange.Severaldustmodelshave
natesthetotaldustmassandthe absorptionin the visibleand
beenproposedtoexplaintheobservations.Recentdustmodels
theNIR.Observationsofthe”silicate”absorptionfeaturenear
share many common characteristics (e.g. Mathis at al. 1977,
10µmindicatethattheirmassisdominatedbyamorphoussil-
Draine & Lee 1984, Draine & Anderson1985, Weiland et al.
icates(seeKemperetal.2004).Suchamorphousstructurefor
1986, De´sert et al. 1990, Li & Greenberg 1997, Dwek et al.
thesedustgrainsisexpectedfromtheinteractionwithcosmic
Sendoffprintrequeststo:[email protected] rays which should alter the nature of the solid (Brucato et al.
2 C.Menyetal.:Far-infraredtomillimeterastrophysicaldustemission
2004,Ja¨geretal.2003),butalsofromtheformationprocesses tionsregardinggraincollisions,destructionandagglomeration,
oftheseBGsbyaggregationofsub-sizedparticles. includingpreplanetarybodiesformationincircumstellardisks.
Recentstudiesattemptingtoexplainthesubmillimeterob- InSect.2,wefirstrecallsomebasicsregardingtheFIR/mm
servations,revealedtheneedforeithertheexistenceofanad- dust emission and the semi-classical model of light interac-
ditionalmostmassivecomponentofverycolddust(e.g.Reach tion with matter. In Sect.3, we gather evidences for spectral
etal.1995,Finkbeineretal.1999,Gallianoetal.2003)orfor variationsinastronomicalandlaboratorymeasurements.Then,
substantialmodificationsofdustpropertiesinthesubmillime- we present in Sect.4 a model, based on the intrinsic proper-
ter with respect to predictions of standard dust models (e.g. tiesofamorphousmaterials,takingintoaccountabsorptionby
Ristorcellietal.1998,Bernardetal.1999,Stepniketal.2003a, DisorderedChargeDistributionaswellastheeffectofabsorp-
Dupacetal.2001,Dupacetal.2002,Dupacetal.2003). tion by a distribution of TLS. In Sect.5, we propose a new
modelofthesubmillimeterandmillimetergrainabsorptionand
Similarly,laboratoryspectroscopicmeasurementsofinter-
discuss the range of plausible values for the free parameters,
stellar grains analogsrevealthat noticeable variationsin their
based on existinglaboratorydata. In Sect.5.4, we discuss the
optical properties in the Far Infra-red (FIR) and Millimeter
implicationsofthemodel,inparticularregardingtheexpected
(mm) wavelength range can occur (Agladze et al. 1994,
variationsof the FIR/mm emission spectra with the dusttem-
Agladzeetal.1996,Mennellaetal.1998,Boudetetal.2005).
perature.Sect.6 is devotedto conclusions.The determination
They pointed out the possible role of two phonons differ-
of the model parameters applicable to astronomical data will
ence processes, disorder-induced one-phonon processes, res-
addressedinacompanionpaper.
onant or relaxation processes in the presence of Two-Level
Systems(TLS)inamorphousdust(seeSect.4.3).Theinfluence
ofsuchTwo-LevelSystemsoninterstellargrainabsorptionand 2. BasicknowledgeontheFIR/mmdustemission
emission properties was first proposed to the astronomers’s
2.1.Interstellardustemissionandextinction
community by Agladze et al. 1994. More recently, a prelim-
inary investigations by Boudet et al. 2002 showed that opti- Theintensityofthermalemissionfrominterstellardustattem-
calresonantandrelaxationtransitionsinadistributionofTLS peratureT is
d
couldqualitativelyexplaintheanticorrelationbetweenthetem-
perature and the spectral index of dust as observed by the Iω(ω,T)=ǫe(ω) Bω(ω,Td) (1)
·
PRONAOSballoonexperiment(e.g.Dupacetal.2003).
whereI isenergyfluxdensityperunitarea,angularfrequency
ω
A precise modeling of the long wavelength dust emission andsolidangle(orspecificintensity),ǫ thedustemissivityand
e
is important in order to accurately subtract foreground emis- B thePlanckfunctionatangularfrequencyω.
ω
sion in cosmological background anisotropy measurements, Accordingto the Kirchhoff law, the emissivity ǫ is equal
e
especially for future missions for space CMB measurements totheabsorptivity
(Lamarre et al. 2003) and for surveys of foregroundcompact
sources(seeGromovetal.2002).Anumberofcomprehensive ǫe(ω)=1 e−τ(ω). (2)
−
foregroundanalysispapers(Bouchet&Gispert1999,Tegmark
where the optical depth τ is related to the dust mass column
etal.2000,Bennettetal.2003,Barreiroetal.2006,Naselsky
densityonthelineofsightM andthedustmassopacityκas
d
et al. 2005and other)did not pay attention to dustemissivity
modelaccuracy.In addition,precise modelingofthe FIR/mm τ(ω)=κ(ω) M (3)
d
·
dust emission is also importantto derive reliable estimates of
Forsphericalgrainsofradiusaanddensityρ,thedustopac-
the dust mass, to trace the structure and density of pre-stellar
ity(effectiveareapermass)isgivenby
cold cores in molecular clouds. Indeed the dust governs the
cloud opacity,and influencesthe size of the cloud fragments.
3 Q(ω)
Inaddition,theefficiencyofstarformationshouldberelatedto κ(ω)= , (4)
4ρ a
thedustproperties,whichareexpectedtovarybetweenthedif-
fusemediumandthedensercoldcores.Itisthusofimportance wheretheabsorptionefficiencyQ(ω)=σ(ω)/(πa2)istheratio
toknowhowsomephysicalorchemicalpropertiesofdustcan of the absorption cross section σ(ω) to the geometrical cross
influencetheirFIR/mmemission. sectionofthegrainπa2.ThegrainequilibriumtemperatureT
d
Formodelsofthesubmillimeterelectromagneticemission isdeducedfromthebalancebetweentheemittedandabsorbed
of dust, consideringthe intrinsic mechanicalvibrationsof the radiationfromtheInter-StellarRadiationField(ISRF)
grainstructureis naturalsince theyfallin the rightfrequency
range(typicallyinthefrequencyrangeν 1011 1013Hz,i.e. ∞B (ω,T )Q(ω)dω= ∞IISRFQ(ω)dω (5)
∼ − ω d ω
λ 30µm 3mm).Observationsinthisspectralregionpoten- Z0 Z0
∼ −
tiallyprovidesnewtoolsforinvestigatingtheinternalstructure The Mie theory,using the Maxwell equationsof the elec-
ofdustgrainmaterial.Theinternalmechanicsofgrainsisalso tromagnetictheory,leadstoanexactsolutionfortheabsorption
important for chemistry in the ISM. The dust vibrations sup- andscatteringprocessesbyanhomogeneoussphericalparticle
plyanenergysinkfornewlyformedmoleculeswhichareusu- ofradiusa whosematerialischaracterizedbyitscomplexdi-
allyformedinunstableexcitedstates.Understandingthegrain electric constant ǫ(ω) = ǫ (ω)+iǫ (ω). In that case, the ab-
′ ′′
structure and intrinsic movementsis necessary for considera- sorption coefficient Q(ω) writes as an infinite series of terms
C.Menyetal.:Far-infraredtomillimeterastrophysicaldustemission 3
related to the complex dielectric constant. For particles small wherek isthecomplexvalueofthewavenumberk =ω√ǫ/c
compared with the wavelength such as √ǫ(ω) 2πa/λ 1 andǫ isthecomplexdielectricconstantThereforeforǫ ǫ ,
′′ ′
| |· ≪ ≪
(whereλ=2πc/ωisthewavelengthandcthespeedoflightin
ωǫ
vacuum),theabsorptioncoefficientcanbeapproximatedby α= ′′ (12)
c√ǫ
′
8πa ǫ(ω) 1
Q(ω)= Im − (6) Industandporousmaterialmeasurements,amassabsorp-
λ · ǫ(ω)+2
" # tioncoefficientα/ρisalsooftenused.
IntheFIR/mmwavelengthrange,fornonconductingmate-
rials of astronomicaldust particles , the conditionǫ′′ ǫ′ is Inthefollowinganalysis(Sect.2.4andother)wewilllink
≪
generally satisfied, and in this case the absorption coefficient themacroscopicvalueǫ tothevalueofthelocalsusceptibility
reducesto χ =χ +iχ definedonmicroscopicscale.
o ′o ′o′
Q(ω) ω 12 Becausesmallastronomicalparticlescannotalwaysbecon-
= ǫ , (7)
a c (ǫ +2)2 ′′ sideredasacontinuousmedium,theelectrodynamicequations
′
ofcontinuousmediamustbeappliedwithcare.Sowedetailin
As a consequence, following Eq.(7) the dust opacity
thefollowingtheequationsused,andwediscusstheirrangeof
rewrites
application.
ω 9 ǫ
κ(ω)= ′′. (8) TheEquationsofmotiondescribestheresponseofcharges
c (ǫ′+2)2 ρ toalocalelectricfieldE0,whichpermitstodetermineadipole
momentperunitvolumeas P = χ E .Itshouldbetakeninto
0 0
2.2.Assumptionofaλ-independentemissivity accountthatthelocalelectricfield E0 differsfromanexternal
spectralindex field E due to additional field produced by other parts of the
dielectric.Inthecaseofisotropicdielectricmaterials,
Intheabsenceofspecificknowledgeaboutrealisticdependen-
4π
cies for ǫ(ω), a simple power law approximationfor FIR/mm E = E+ P.
0
dustemissionisoftenassumed 3
ω β ThedielectricconstantǫlinkstheexternalfieldEandtheelec-
κ(ω)=κ(ω0) ω , (9) tric inductance D = E+4πP through D = ǫE. In this case,
0! ǫ canbederivedfromthesusceptibilityχ ,usingthesocalled
0
whereβisreferredtoastheemissivityspectralindex. Clausius-Mossottiequation
The intensity of thermal emission from interstellar dust,
4πχ
assuming an optically thin medium, at a given wavelength ǫ =1+ 0 . (13)
λ=2π/ωlocatedintheFIR/mmrangeis 1− 34πχ0
ω β Whenǫ′′ ǫ′,wehave
I (ω,T)=ǫ (λ ) B (ω,T ) (10) ≪
ω e 0 · ω0! · ω d ǫ (ǫ′+2)2 4πχ , (14)
where ǫ (λ ) is the dust emissivity at a reference wavelength ′′ ≈ 9 ′o′
e 0
λ =2πc/ω .
0 0 whichleadsto
Simple semi-classical models of absorption such as the
Lorentz model for damping oscillators and the Drude model
forfreechargecarriers(Sect.2.4)provideanasymptoticvalue α(ω) = 4πω (ǫ′+2)2 χ (ω), (15)
β = 2 when ω 0. Such a value of the spectral index was c√ǫ 9 ′o′
→ ′
in satisfactory agreementwith the earliest observationsof the ω4πχ
κ(ω) = ′o′. (16)
FIR/mmSEDofinterstellardustemission,butnotnecessarily c ρ
withthemostrecentones.
The emissivity spectral index β is equal to the slope of a In Eq.(15), the term (ǫ′ + 2)2/9 is the local field correction
plotofthedustopacityversuswavelengthinlogarithmicscale. factor,theterm √ǫ istherefractionindex.
′
Inthegeneralcase,thepower-lawassumptionisnotvalid,and Thus,therelationbetweentheabsorptioncoefficientαde-
the spectral index has to be considered as a wavelength and duced from transmission measurements in the laboratory and
temperaturedependentparameter.Similarly,ǫ (λ )inEq.(10) the absorption efficiency Q or opacity κ, for small spherical
e 0
shouldbeconsideredafunctionofT. grainsofradiusaanddensityρcanbeobtainedfromequations
Eq.7and12,andwrites
2.3.Relationsbetweenabsorptionpropertiesofbulk Q(ω) 12a √ǫ
= · ′ (17)
materialanddust α(ω) (ǫ +2)2
′
The absorption coefficient defined as the optical depth in the andasaconsequence
bulkmaterialperunitlengthis
κ(ω) 9 √ǫ
α=2Im(k), (11) α(ω) = ρ (ǫ· +2′)2 (18)
· ′
4 C.Menyetal.:Far-infraredtomillimeterastrophysicaldustemission
AsolutionofEq.(19)withE(t)= Eˆe iωt isu=uˆe iωt,
− −
The condition ǫ ǫ is valid for dust materials in the
FIR/mm wavelength′′ra≪nge,′implying that the real part of the uˆ = qjEˆ ·e∗κjω2κ . (24)
dielectricconstantǫ′ canbeconsideredtofirstorderasacon- Xκ j ω2κ −ω2−iγκω
stant, leading to dǫ /dω = 0. This comes from the coupling
′ Dielectric propertiesof a substance can be describedby a
between the real part ǫ and the imaginary part ǫ through
′ ′′ susceptibilityχ ,whichisthedipolemomentperunitvolume
theKramers-KronigrelationsexposedinSect.3.2.Thisimplies 0
generatedbyaunitylocalstrenght Eˆ =1electricalfield,
that,inthatspectralrange,theabsorptioncoefficientα,theab- | |
sorptionefficiencyQandtheopacityκaresimplyproportional. ρ ω2
χ = κ κ . (25)
Thustheslopesofα(ω),Q(ω)andκ(ω)arethesame.Asforthe o ω2 ω2 iγ ω
emissivity,anabsorptivityspectralindexcanthenbedefinedby Xκ κ − − κ
theslopeofthespectralplotoftheabsorptionα(ω)inlogarith- Forisotropicmaterials,constants
micscale.Thisabsorptivityspectralindexdeducedfromlabo-
ρ = q q e (Eˆ e ) (26)
ratorytransmissionmeasurementscanbedirectlycomparedto κ j′ j κj′ · ∗κj
theemissivityspectralindexdeducedfromastrophysicalmea- (cid:12)Xj j′ (cid:12)
(cid:12) (cid:12)
surements. does(cid:12)(cid:12)notdependuponthe(cid:12)(cid:12)directionofEˆ.
In the extreme case where ω γ the spectrum of the
κ κ
≫
dielectric absorption can be described as a collection of non-
2.4.Thesemi-classicalmodels
overlappingresonantLorentzprofiles
Thewellknownsemi-classicalmodeloflightinteractionwith
ρ ω2γ ω
chargecarriersinmatter(electronsandions)considersthemo- χ = κ κ κ . (27)
′o′ (ω2 ω2)2+(γ ω)2
tionofchargedparticlesdrivenbythetime-dependentelectric κ − κ
field of a monochromatic plane wave, magnetic forces being It is the case of infrared absorption bands, especially in
neglectedcomparedwithelectricalforces. ionic crystals (”reststralen” bands). This corresponds to the
Theequationsofmotionofinteractingchargecarriersare: thirdtermofEq.(19)beingsmall.
Intheoppositecase(ω γ )thisleadstoaDebyerelax-
κ κ
≪
ationspectrumwhenω ω :
m u¨ξ+ Kξξ′uξ′ + bξξ′u˙ξ′ =q Eξ(t), (19) ≪ κ
j j jj j jj j j
′ ′ ′ ′ ρ (ωτ )
Xj′ξ′ Xj′ξ′ χ′o′ = 1+κ (ωτκ)2, (28)
κ
wherem ,q aremassesandchargesofparticles,Kξξ′ arecon-
stantsofjelasjticinteraction,Eξ isξ-thcomponentojfj′electrical whereτκ = γκ/ω2κ isa relaxationconstant.Thisis usuallythe
caseofpolardielectricliquids.Itcorrespondstothefirstterm
field vector E, uξ are components of the 3-dimensional vec-
j ofEq.(19)beingnegligible.
torof j-thiondisplacement.Dampingparametersb describes
j Thisspectrumcorrespondsto situationswhere the inertial
dissipationofenergy.ForceconstantsK arethecoefficientsof
effects (first term in Eq.19) are comparatively small. If this
powerexpansionofthepotentialenergyU.
condition is satisfied, the time constant in Eq.(28) can usu-
ally be determined by measuring or calculating the timescale
1 of energydissipation duringreturn to equilibriumstate, with-
U = Kξξ′uξuξ′ +... (20)
2 jj′ j j′ outsolvingEq.(19).
jξ
X′ ′ Athirdextremecase,correspondingtothesecondtermof
U( uξ ) dependsonlyon relativedeviations(uξ uξ) Eq.(28) being negligible is usually refered to as the Drude
··· j··· j − j′ approximation.It applies to free carriers and is considered in
withcorrespondingsymmetryofKξξ′,particularly
jj Sect.2.6.
′
Kξξ′ = Kξ′ξ, Kξξ′ =0. (21) 2.5.TheLorentzmodelforcrystallineinsulators
jj j j jj
′ ′ ′
jξ
X′ ′ The Lorentz model applies for crystalline isolators with an
LinearEq.(19)with E = 0havesolutionsintheformofa ionic character, and describes the resonant interaction of an
sumofdampingoscillations electromagneticwavewiththevibrationsofionswhichactlike
bound charges. Such an interaction is caused by the electric
u= aκeκ exp( γκt+iωκt), (22) dipoles which arise when positive and negative ions move in
· −
oppositedirectionintheirlocalelectricfield.Thiskindofdust
X
where
absorption can be clearly identified in astrophysical observa-
u=( uξ )isthemultidimensionaldeviationvector,a
··· κj··· κ tionsasemissionorabsorptionbands.Indeed,theopticaldepth
areconstants,eκ =(···eκξj···)areeigenvectors,andγκ−iωκare τisproportionaltoopacityκ(Eq.16):
eigenvaluesofmatrix
4π ρ ω2γ ω2
κ(ω)= κ κ κ (29)
k−mjδjj′δξξ′ +Kξjξj′′ +ibξjξj′′k. (23) cρ Xκ (ω2κ −ω2)2+(γκω)2
C.Menyetal.:Far-infraredtomillimeterastrophysicaldustemission 5
The maximum number of optically active vibrational However the optical behavior of amorphous materials is
modes in a crystal is n = 3(s 1), where s is the number different. The disorder of the atomic arrangement leads to a
opt
−
ofatomsin anelementarycellofthe lattice.Theyare usually breakdown of the selection rules for the wavenumber. The
classified as branches in dispersion dependenceof oscillation absorption spectra reflects thus the whole density of vibra-
frequencyω(k) on wave numberk. All these branchesare re- tional states. This induces in the longest wavelength range a
strictedbyrelativelyhighω(k)intheinfraredregion.Another3 broad absorption band due to single phonon processes which
brancheswithω(k) 0whenk 0areacousticmodes.They completely dominates over the multiphonon absorption (H.
→ →
areopticallynotactiveincrystalbecauseatotalchargeofcell HenningandH.Mutschke1997).Asaconsequence,suchlat-
is zero. Anothersituation for amorphousmatter is considered ticeabsorptiondoesnotvanishatlowtemperaturefortheamor-
hereinSect.4.2. phoussolids,incontrasttothecrystallineones.
For substances of astrophysical interest, the characteristic
frequenciesofbandsfallinregionλ<50µm.Therefore,inthe
2.6.TheDrudemodelforconductingmaterials
FIR/mmwavelengthrange,the asym∼ptoticω 0 ofEq.(29)
→
canbeused: For free charge carriers the previous semi-classical equa-
tion of motion (Eq.19) can still be used, setting all restor-
4π ρ γ
κ(ω)=ω2 κ κ (30) ing force coefficients K to zero, equaling all the masses
cρ κ ω2κ mj to the effective mass of the free carriers mc ( mj =
X
m ), and using a constant value for the damping parameters
c
Thus, the Lorentz model formally applied to dust in the b /m =const=γ .ThereforeEq.(25)forthesusceptibilitysim-
j j c
FIR/mmwavelengthrangegivesaspectralindexβ=2,avalue plifies as χ = ω2 (ω2 iγ ω) 1, where ω is the so-called
oftenusedin”standard”dustmodels. plasmafreqouen−cyωp2 = N−q2/cm −,andN isthpenumberoffree
p c c c c
Manydustmodelshaverelatedthelongwavelengthabsorp- carriersperunitvolume.
tionbyinterstellarsilicatesandices(λ>100µm)tothedamp- In metals, the numberof free electronsper unitvolume is
ingwingoftheinfrared-activefundamentalvibrationalbands, independentof temperature,andis of the same orderofmag-
visibleatshorterwaves(seeforexampleAndriesseetal.1974 nitudethanthenumberofatoms(N 1022cm 3).Thisleads
c −
andreferencestherein). to a frequency ω 1015s 1 locate∼d in ultraviolet or in the
p −
Laboratory infrared data are satisfactorily described (see visible,andtocorre∼spondinglyhighconductivityσ=ω2/γ .
p c
forexampleRast1968)bya sumofresonantLorentzprofiles In the long wavelength range, far from the plasma fre-
(or reststrahlen bands) as described by Eq.(25) and Eq.(29), quency(ω ω )andwhenthemodulusofǫ islargecompare
p
inwhichthevariousconstantsρκ, ωκ, γκ aredeterminedfrom to1,theasy≪mptoticbehavioroftheopacityisgivenby
spectroscopic measurements. This model gives a satisfactory
9γ2
approximationfortheinfraredregion,notonlyforioniccrys- κ(ω)=ω2 c , (31)
tals, butalsoforcovalentonesandfortheir amorphouscoun- cρω4
p
terpartswithanappropriatechoiceofρ ,ω ,γ .
κ κ κ whichalsogivesaspectralindexβ=2.Metallicmaterialsand
The situation is more complicated for the low frequency
high conductivity semiconductors are expected to follow this
region,whereabsorptionisdefinedbydampingconstantγ,in
model.Possible dustabsorbentsofsuchtypearegraphiteand
contrasttoIRbands,whereabsorptionisweaklydependenton
metalliciron.
γ. The damping constants b introduced in the semi-classical
j Therefore, both the Lorentz and the Drude models in the
model (Eq.19) and therefore γ in Eq.(29) have no simple
κ low frequencylimit yieldβ=2 in Eq.(9). These very different
microscopicexplanation.The dampingis linked to heating or
modelshavecommonfeatures.Theircharacteristictimes(ω 1,
distribution of energy to all modes of grain vibrations. So γ −κ
ω 1)arefarfromFIR/mmregion.Inthiscaseagoodapprox-
characterizes intermode interactions which are frequency and −p
imationisobtainedbyasymptoticexpansionoftheabsorption
temperaturedependentingeneral.Incontrasttosinglephonon
coefficientα(ω)onevenpowersoffrequencyω2n,n>0.Inthe
absorptionofresonantIRphotons,awingabsorptionisatwo
generalcase the first term is dominating(n = 1) and then we
phonondifferenceprocess.Duetotheconservationofenergy,
haveβ=2.
the energy of the absorbed low frequency photon is equal to
Othervalues(e.g.n=2,β=4)arepossibleifphysicalmech-
the energy difference of the two phonons in the fundamen-
anisms suppressingthe n = 1 term are present, as in the case
tal vibrational bands (Rubens and Hertz 1912, Ewald 1922).
of screening considered in Sect.4.2. Intermediate values of β
In fact in crystals, any combination of acoustical and optical
provideaclearevidenceforspecificprocesseswhichtimecon-
phononswhich satisfies the energy and momentumconserva-
stantsareoftheorderofω 1.
tion rules can take part to the absorption . This is the main −
source of infrared absorption in homoatomic crystals. In the
FIRwavelengthrangethetwo-phonondifferenceprocessesare 3. Temperatureandspectralvariationsofdust
responsiblefortheabsorptioninioniccrystals.Becausetheef- opticalproperties
ficiency of such processes are entirely determined by the dif-
3.1.Astronomicalevidences
ferencebetweenthephononoccupationnumbersofthetwoin-
volvedphonons,thecorrespondingabsorptiondecreasesdras- Analyses of millimeter and submillimeter emission from
ticallyatdecreasingtemperatures(Hadni1970). molecular clouds have found spectral indices between β=1.5
6 C.Menyetal.:Far-infraredtomillimeterastrophysicaldustemission
(Walker,Adams&Lada,1990)andβ=2(Gordon1988,Wright surementsintheIRandFIRevidencedastrongmillimeterex-
etal.,1992).Howeversomevaluesin excessof 2canalso be cess towardsa set of fourblue compactgalaxy(NGC1569,II
found in the literature (Schwartz 1982, Gordon 1990). A few Zw40,He2-10andNGC1140).Theyattributedthisexcessto
studies of dense clouds have yielded spectral indices around thepresenceofverycolddust(T = 5 7K)witha veryflat
d
−
1 (Gordon1988,Woodyetal. 1989,Oldhametal. 1994),but emissivity index (β = 1) which would have to be concentrat-
observationsof disks of gas and dust around young stars can ingintoverydenseclumpsspreadalloverthegalaxy.Thisvery
indicate values sometimes less than 1 (Chandler et al. 1995). colddustwouldthenaccountforabouthalformoreofthetotal
A submillimetercontinuumstudyofthe Oph1 cloud(Andre, dustmassofthosegalaxies.
Ward-Thompson& Barsony 1993)found a typical value of β
equal to 1.5 but attributed changes in observed ratios to tem-
3.2.Laboratoryevidences
peraturevariations.Withoutappropriatetemperaturedata, the
authorscouldnotbeconclusiveonthisissue. 3.2.1. Comparinglaboratoryandastronomicaldata
More recent observations have evidenced that the actual
As it was shown in Sect.2.3, laboratory data on absorption
spectral energydistribution of dust emission could be signifi-
spectralindexcouldbeusedforinterpretingastronomicalob-
cantlymorecomplicatedthandescribedabove.Analysisofthe
servations,takingintoaccountthequantitativelimitationscon-
FIRAS results have shown that, along the galactic plane, the
sideredhere.Someotherlimitationsshouldbeconsidered.
emissionspectrumissignificantlyflatterthanexpected,witha
First,thereisadifferencebetweenbulklaboratorysamples
slope roughly compatible with β = 1 (see Reach et al. 1995,
andthesmallcosmicdust.Thelatterprobablyhasahighsur-
in particular their Fig. 7). They attributed the flattening com-
facetovolumeratio,andsurfaceeffectsareimportantforspec-
paredtotheβ = 2canonicalvaluetoanadditionalcomponent
troscopyasforthephysicsandthechemistryofdust.
peakingin the millimeter range and favored that the millime-
Second, samples used for laboratory measurements are
ter excess could be due to the existence of very cold dust at
generally synthesized in very small amounts. Along with the
T = 5 7K inourGalaxy.However,theoriginofsuchvery
d
− small values of the absorption coefficients it requires high-
cold dust remainedunexplained,and the poor angularresolu-
sensitivity dedicatedinstrumentssuch as 3He-cooledbolome-
tionoftheFIRAS data( 7 )didnotallowfurtherinvestiga-
◦
≃ tersor/andhighpowersourcesofradiation.
tions.
In the laboratory indirect IR/FIR/mm spectroscopy meth-
Finkbeineretal.1999lateradoptedasimilarmodel.They
ods are also used, such as Kramers-Kronigspectroscopy.The
showedthatagoodfitoftheoverallFIRASdatacouldbeob-
dielectric constant counterparts (ǫ 1) and ǫ are Hilbert
tainedusingamixtureofwarmdustatT 16Kandverycold ′ ′′
d −
≃ transformsofeachotherandarerelatedthroughtheKramers-
dustatT 9K.Thetemperatureofthewarmcomponentwas
d
set using a≃fit to the DIRBE data near λ = 200µm while the Kronigrelations(seeLandau&Lifshits1982)by
temperatureof the very-coldcomponentwas set assuming an
independent dust component immersed in the same radiation
1 ǫ (x)
fieldbutwithdifferentabsorptionproperties.Theyimpliedthat ǫ′(ω) = ∞ ′′ dx+1,
π x ω
these two componentscould be graphite (warm) and silicates Z−∞ −
(32)
(very cold). However, they gave no quantitative argument to
1 ǫ (x)
supportthishypothesis ǫ (ω) = ∞ ′ dx,
′′
−π x ω
PRONAOS was a french balloon-borne experiment de-
Z−∞ −
signed to determine both the FIR dust emissivity spectral in-
wherebothintegralsaretheCauchyprincipalvalues.
dex and temperature, by measuring the dust emission in four
AsimpleformofEq.(32)isforω=0:
broadspectralbandscenteredat200,260,360and580µm(see
Lamarre et al.1994, Serra at al.2002, Pajot et al.2006). The 2
ǫ (0)= ∞ǫ (x)dlnx+1. (33)
analysis of the PRONAOS observations towards several re- ′ π ′′
gionsoftheskyrangingfromdiffusemolecularcloudstostar Z0
formingregionssuchasOrionorM17haveevidencedananti- Theuseoftheserelationstogetherwithreflectionmeasure-
correlation between the dust temperature and the emissivity ments R(ω)= (√ǫ 1)/(√ǫ+1)2 allows to resolve the sys-
| − |
spectral index (see Dupac et al. 2003). The dust emissivity temofequationwithrespecttoǫ (ω)andǫ (ω)andtherefore
′ ′′
spectralindexvariessmoothlyfromhighvalues(β 2.5)for to calculate an absorption spectrum. Possible method inaccu-
≃
colddustatT 12K tomuchlowervalues(β = 1)fordust racies due to inverse problem solution should be considered
d
≃
athighertemperature(uptoT 80K)instarformingregions carefully.Inparticular,itissometimesassumedasapriorthat
d
≃
like Orion. Dupac et al. (2003) showed that these variations thesolutionhastheformofapowerspectrum.Inpractice,the
werenotcausedbythefitprocedureusedandwereunlikelyto most reliable approach should be to use the same model for
beduetothepresenceofverycolddust.Theyconcludedthat interpretationofboththeastrophysicalandlaboratorydata.
thesevariationsmaybeintrinsicpropertiesofthedust.
Extragalactic observations have also revealed unexpected
3.2.2. Spectralindexvariationsindisorderedsolids
behavior of dust emission at long wavelength. For instance,
Gallianoetal.(2003,2005),combiningJCMTdataat450and Ithasbeenknownformorethan25years,thatdisorderedsolids
850µm,IRAM30mdataat1.2mmwithISOandIRASmea- (glassesand mixedcrystals)have significantlyhighersubmil-
C.Menyetal.:Far-infraredtomillimeterastrophysicaldustemission 7
limeterabsorptionthanperfectcrystalsofsimilarchemicalna- range of temperature and revealed a variation of the spectral
ture. behavioroftheabsorptioncoefficientwithtemperature.
Monetal.(1975)performedfirstabsorptionmeasurements Agladzeetal.(1996)performedabsorptionmeasurements
ontypicalinterstellaranaloggrains(crystallineandamorphous
on various amorphous dielectric materials in the millimeter
silicategrains)atlowtemperature(1.2-30K)inthemillimeter
wavelength range (1mm - 5mm) and at low temperature (in
the range 0.5K - 10K). A strong temperature-dependenceof range (700 µm- 2.9mm). They found an unusual behavior of
the millimeterabsorptionofamorphoussilicates :the absorp-
theabsorptionwasobserved,leadingtoanmillimeterabsorp-
tionexcessatlowtemperature.Theyalreadyattributedthisef- tioncoefficientdecreaseswithtemperaturedowntoabout20K
andthenincreasesagainwithdecreasingtemperature.Theyde-
fect to the presence of a distribution of Two-levelSystems in
scribedthefrequencydependenceoftheabsorptioncoefficient
thestudiedmaterials.
using a temperature-dependent spectral index. Depending on
Others experimental results on bulk materials at frequen-
the samples, the spectral index can decrease from β = 2.6 at
ciesω/2πcbetween0.1and150cm 1andT between10Kand
− 10K down to β = 1.8 near 30K. They referred to tunneling
300K were discussedby Strom& Taylor(1977). The spectra
effectinTwoLevelsSystemstoexplainthisbehavior.
follow a generalempiricalrelation α = Kω2 between 15 and
Mennella et al. (1998) measured the temperature depen-
150cm 1, where the temperature independentconstant K de-
− dence of the absorption coefficients between 295K and 24K
pendsonthematerialconsideredanditsinternalstructure.At
and for wavelength between 20µm and 2mm on different
lower frequenciesω/2π <15cm 1 (λ > 700µm) the slope of
− kindofcosmicgrainanalogs(amorphouscarbongrains,amor-
thespectraisvariable(βbetween0and3)andtemperaturede-
phous and crystalline silicates ). They reported a significant
pendentasshowninStrom&Taylor(1977).
temperature-dependenceofthespectralindexoftheabsorption
Bo¨sch(1978)performedabsorptionmeasurementsfortem- coefficient,particularlystrongfortheiramorphousiron-silicate
perature between 300K and 1.6K in the FIR/mm range sample.Theirmeasurementsshoweda systematicdecreaseof
(500µm - 10mm) on amorphous silicates mainly composed thespectralindexwithincreasingtemperature.
of SiO2. The studied silicate is not a typical grain analog. Boudetetal.(2005)performedmeasurementsondifferent
However,measurementsperformedoversuch a wide spectral types of amorphous silicates (typical analog grains and SiO
2
and temperature range are particularly appropriated to study samples) for temperature between 10K and 300K and wave-
physicsgoverningabsorptionprocesses.Bo¨schfoundastrong lengthfrom100µmto2mm.Theyfoundastrongtemperature
temperatureandfrequencydependencyoftheabsorptioncoef- andfrequencydependencyofthe absorptioncoefficient.They
ficient in the millimeter range (λ > 500µm). The absorption definedtwospectralindexdependingonthewavelengthrange
coefficientwascharacterizedbyanincreaseofthespectralin- considered. For wavelengths between 500µm and 1mm they
dexwith temperaturefromβ = 1.6at300K to β = 3 at10K foundapronounceddecreaseofthespectralindexwithincreas-
andbyastrongtemperaturedependence.Todescribethistem- ingtemperaturewhereas,forwavelengthsbetween100µmand
peratureandfrequencybehavior,Bo¨schalsoreferredtotheex- 200µm,thespectralindexwasfoundtobeconstantwithtem-
istenceofTwoLevelSystemsinthematerial. perature. They put some SiO sample through a strong ther-
2
Therefore,laboratoryspectroscopyshowsevidencesforde- maltreatmenttoremovemostSiOHgroups,andobservedthat
viationofabsorptionspectralindexfromthecanonicalβ=2in thetemperature-dependentabsorptiondisappearstotally.They
theFIR/mm.Thetemperatureindependentabsorptionprocess thus identified the silanol groups as the defects that, in their
whichprovidesβ = 2 inthe spectralregionλ < 700µmdoes silicate sample, are at the origin of the behavior. Considering
notseemtodominatetheabsorptionatlongerwavelengths.We that the OH groups can not simply increase the coupling be-
will see in Sect. 4 that the model proposedhere includes this tweenthephotonandthebulkphonons,theyassumedreason-
phenomenoninanaturalmanner. able that the defects induce closely spaced local energy min-
ima, which correspond to the dynamics of a ”particle” in an
asymetricdouble-wellpotential.
3.2.3. Spectralvariationsindustanalogues Theselaboratorystudiesprovidestrongindicationsthatthe
absorption coefficients of amorphous grain analog materials
FIR/mmopticalpropertiesofsolidswithchemicalcomposition
varysubstantiallybothwithtemperatureandfrequency.
andstructureconsideredasanaloguesofinterstellardustwere
investigated in a few laboratory studies (for example Koike
et al. 1980, Agladze 1996), Henning and Mutschke 1997, 4. Amodeltoexplaintemperatureandspectral
Mennellaetal.1998,Boudetetal.2005).Particularinteresthas variationofdustopticalproperties
beengiventolowtemperatureinvestigationsofamorphousor
4.1.Modelingdisorderedstructure
otherdisorderedmaterials,whichrevealsystematicdeviations
from the phonon theory of crystal solids and was interpreted
Two mechanisms have to be considered when dealing with
usingtheTLStheory(seeBo¨sch1978,Phillips1987andKu¨hn amorphousmaterials,inordertotakedisorderintoaccount.
2003).
First, one should consider the acoustic oscillation excita-
A few laboratory measurements have been performed on tionbasedupontheinteractionofsolidswith electromagnetic
amorphous solids in the FIR/mm range at low temperature. wavesduetoDisorderedChargeDistribution(SeeSchlo¨mann
However, some of these studies were performed over a wide 1964). This effect takes place not only in amorphous media,
8 C.Menyetal.:Far-infraredtomillimeterastrophysicaldustemission
but also in disordered crystals like mixed and polycrystals, spondstoEq.(19)withb = 0.Dissipationeffectsariseinthe
j
andinsomemonocrystalswithinversespinelstructure,forex- phononconceptastheresultofphononinteraction,takinginto
ample.Thisabsorptionis temperatureindependent.TheDCD accountanharmonicterms in Eq.(20). Their role was pointed
model introduces a correlation length l which quantifies the firstbyDebye(1914).
c
scale over which the atomic structure of the material realizes For harmonic oscillations u = uˆe iωt with electrical field
−
chargeneutrality.Theabsorptionspectrumofsuchastructure E(t)= Eˆe iωt Eq.(19)takestheform
−
presentstwoasymptoticbehaviors.Towardsshortwavelengths,
q
theabsorptionischaracterizedbyaspectralindexβ=2andin ω2uˆξ+ Dξξ′ uˆξ′ = j Eˆξ, (34)
thelongestwavelengthrange,thespectralindextendstowards − j jξ jj′ j′ mj
X′ ′
β=4.Thelocationofthespectralrangeinwhichthetransition
wherethecoefficientsofthedynamicsmatrixare
between the two regimes is directly related to the correlation
Dξξ′ = Kξξ′/m .
length.TheDCDmodelisdescribedinSect.4.2. jj jj j
′ ′
The second mechanism describes the disorder at atomic
Thesolutionofthedispersionequation
scale, as a distribution of tunneling states, which leads to a
temperature-dependentoptical absorption, and enables to ex- ω2eξ+Dξξ′ eξ′ =0 (35)
plain the temperature dependence of the spectral index ob- − j jj′ j′
served in laboratory experiments. This model was originally leadsto resonantfrequenciesωκ andcorrespondingeigenvec-
elaborated to explain the unusual thermal and optical proper- tors eκ = (···eκξj···). In an orthonormal basis of vectors eκ,
ties of amorphousmaterial at low temperatures and has been when
dVeasrcmriabe(1d9i7n2d)e(tsaeielsablsyoPPhhilillilpipss(11997827)f,oArnadreervsoienw,)H.aInlpeprairnti&c- eξκje∗κ′ξj =δκκ′, (36)
ξ j
ular, the TLS model was developed to explain the fact that, X
atlowtemperatures,thespecificheatofamorphoussolidsex- theequation(34)hasasimpleform
ceedswhatisexpectedfromtheDebyetheoryforsolids.This
(ω2 ω2)uˆ = g Eˆ, (37)
anomalousbehavioriscommontoallamorphousmaterialsand κ − κ κ·
thereforeappearsindependentoftheirdetailedchemicalcom- whereuˆ areamplitudesofnormaloscillations,
κ
positionandstructure.Theexistenceofsuchtwolevelssystems
hasbeenpointedoutbymeansofmicrowaveandsubmillime- gξκ = qjm−j1e∗κξj,
terspectroscopyexperiments(e.g.Agladzeetal.1996,Bo¨sch Xj
1978 and reference therein). The TLS model is described in and denotescomplexconjugates.
∗
Sect.4.3. ThesusceptibilitycorrespondingtoEq.(37)isgivenbythe
Thetwomechanismsconsideredherefortheabsorptionby tensor
interstellardustprobablydominateinthe FIR/mmand longer
wavelengthsdomain.Both havea largedegreeof universality χξξ′ = qj′qjm−j1e∗κξjeξκ′j′ , (38)
withoutspecific signaturescharacterizingthe chemicalnature 0 ω2 ω2 iωγ
of the absorbing substance. On the other hand, they are sen- Xκj′j κ − − κ
sitive to the internal structure of the solids, in particular their orinthemacroscopicallyisotropiccasebythescalar
degreeofordering,andtomechanicalpropertiessuchasden-
sityandelasticity. χ0 =(χ0xx+χy0y+χz0z)/3
andtherefore,
4.2.DCDabsorption
4.2.1. DCDTheory χ′0′ =Imχ0 = 31 (ωωγ2κqjω′q2j)m2−+j1e(ω∗κξjγeξκ)j′2. (39)
κXj′jξ κ − κ
In amorphousmaterials, the lack of long-range order permits
singlephonon/photoninteractionswithexcitationofanymodes Dampingfactorsγκ were introducedhere to get a station-
arysolution.γ valuesdependsonphononinteractions,asdis-
of mechanical vibrations. Low frequency vibrations are not κ
cussedabove.IntheFIR/mmregion,overlappingofresonance
linked to molecular vibrational bands like is the case for ice
curvesproducesa quasi continuousabsorptionspectrum.The
or silicate bands. In infinite media they correspond to travel-
integratedspectruminEq.(39)canbesimplifiedreplacingthe
ingacousticwaves.Infinitebodieslikeinterstellardustgrains,
integral of the sum of weak individualprofiles by the sum of
thesemodescanbedescribedasstandingwaves.Soweusethe
theirintegratedvalues.
term”phonon”hereforquantumofvibrationalmotionnotre-
strictingit(ifnotstatedotherwise)toperiodiclatticeorinfinite + dω π/2
media. ∞Im = . (40)
Theconceptofphononquasiparticles(Tamm,1930)arises Z−∞ ω2κ −ω2−iωγκ ω2κ −γκ2
amsotnhiecraepspurlotxoifmqautiaonnt,izwahtiiocnhotafkveisbirnattoioancaclomunottioonnlyinquthaedrhaatirc- χ (ω) = π qpj′qjm−j1e∗κξjeξκj′, (41)
terms(onatom displacements)ofenergyin Eq.(20). Itcorre- ′0′ 6ω ∆ω
D E |ω−ωXκ|<∆ω/2
C.Menyetal.:Far-infraredtomillimeterastrophysicaldustemission 9
where∆ωisanaveragingintervalsatifying Here designate an ensemble average, ∆r= r r .
j j
h i − ′
Parametersq2, q2 are describedbelow anddo notdependon
ω ∆ω γ . (42) 1 m
≫ ≫ κ ∆r.
Thecorrelationfunctioncanbedeterminedfromaphysical
Anumberofstatesisproportionalto∆ω,thereforethere-
modelofconsideredmaterial.Thefirst,simplestmodelhasan
sultdoesnotdependon∆ωandγ whenconditioninEq.(42)
κ
uncorrelatedchargedistributionandthereforeadelta-function
issatisfied, whichthereforealso definesthe validityregionof
correlation
theresultsofVinogradov(1960)andSchlo¨mann(1964).
In macroscopicallyuniformandisotropiccases the eigen-
φ(∆r)=δ(∆r)=δ(∆x)δ(∆y)δ(∆z). (49)
vectorsofEq.(35)intheformofplanewavesare
Ingeneralcaseacross-correlationfunction,Eq.(49)gives
eξκj =e˜kξηexp(ik·rj). (43) aflatspatialspectrumwithintensitynotdependingonk:
Themanyfoldofsolutionsofthedispersionequationisor- q2
=q2, (50)
dered here in accordance with vector |k| and splits into sev- m k m
eralbrancheslabeledherebyindexη.Eachbranchhasadiffer- (cid:28) (cid:29)
andtherefore
entdispersioncurveω = ω (k).Mostbranchesarerestricted
η
| | ω
to high frequencies(infrared)and only three branches extend χ (ω)= q2 (51)
down to the FIR/mm region. They are so called ”acoustical” ′0′ 12πυ3 m
t
modes. For such low frequencies, these have a linear disper-
Here,weomittedthesecondterminEq.(46)sinceυ 3 2υ 3.
siondependance −l ≪ −t
Anautocorrelationfunctionshouldbeusedform=const
ωη =υηk, (44) φ (∆r)= q(r )q(r) (q2) 1, (52)
a h j i i 1 −
where k = k, and υ is speed of sound. One branch corre-
sponds to lo|ng|itudinalηpropagation (η = l), the two others to withaflatspatialspectrumhq2ik =q21forδcorrelation(Eq.49)
leadingtoEq.(51)withaconstantq2 =q2/m.
2 polarizationsof transverse propagation(η = t). Numericaly, m 1
Inperfectcrystalsthechargedistributionisentirelycorre-
υ < υ. Displacementsofneighborionsin soundoscillations
t l
are synchronous and e˜ξ does not depend on j. The vectors lated forlong distances(long rangeorder)and the absorption
kη mechanism described by Eq.(46) and Eq.(51) does not take
of displacements e = [e˜ x,e˜y,e˜z] are oriented such that
kη kη kη kη place. Nevertheless even for crystals with a perfect lattice, a
e k, e k.Theirabsolutevaluesaredefinedbythenor-
kl k kt ⊥ disorderedchargedistributionispossibleif thelattice permits
malizationequationEq.(36).Thenumberofnormaloscillation
anonuniqueconfigurationofthechargesinanelementarycell,
perunitvolumeinthefrequencyinterval∆ωisN ∆ω,where
ω as is the case, for example of the cubic lattice of salt (NaCl).
ω2 Insuchcases,onlystochasticallydistributedchargesshouldbe
Nω = 4π2υ3η (45) includedinthecalculationofq21,whichcanthenbedefinedas
andtherefore 1 N1 1 N1
q2 = (∆q )2 < q2, (53)
1 N j N j
ωN 2 q2 1 q2 1 j=1 1 j=1
χ (ω)= + , (46) X X
′0′ 24πυ3t *m+ω/υt υ3l *m+ω/υl wheresummingistobedoneonlyforionsinvolvedinthedis-
whereN isnumberofionsperunitvolumeand ordereddistributionand∆q denotesdeviationsfromthemean
chargevalue,whichisdifferentfordifferentnodesofthecrystal
lattice.Theorderedpartofthechargedistributionislinkedonly
q2 1 N q q
m = N2 mj j′ exp(ik·(rj−rj′)). (47) to the spectralcomponentswith ω>υη/d, where d is the lat-
* +k j,j=1 j tice period.This frequencyrangecorrespondsto mid infrared
X′
absorptionbandsandarenotconsideeredhere.
The equation derived here is a generalization of the
In amorphous media the corresponding coefficient q2
Schlo¨mann’s(1964)expressionforonetypeofdisorderedions m
shouldalso excludethe regularpartof the chargedistribution
in a perfect crystal q2 m 1, which was defined by a spatial
h ik − producedbytheshortrangeorderinthemedium.Anequation
powerspectrumofchargedistribution q2 only.
h ik similar to Eq.(53) cannot be written in the general case. The
effectofthisshortrangescreeningcanbeexpressedbythein-
4.2.2. SpectrumofDCDabsorption equality
TheabsorptionspectrumcorrespondingtoEq.(47)canbecal- 1 N q2
culated using correlation functions. It is a cross-correlation q2m < N mj,
functionforgeneralcase j=1 j
X
φ (∆r)= qjqj′ q2 −1 (48) ThesecondmodelofSchlo¨mann(1964)canbeinterpreted
c m m asscreeningoverlargescales.Large-scaleelectricalneutrality
j
(cid:28) (cid:29)(cid:16) (cid:17)
10 C.Menyetal.:Far-infraredtomillimeterastrophysicaldustemission
ofcrystalsandglassescouldbearesultofhighchargemobility
in solutions or melts from which the correspondingmaterials
weremanufactured.
The corresponding correlation function should be written
ingeneralizedcaseas
1
φ(∆r)=δ(∆r) exp(∆r/l ), (54)
− 8πl3 | | c
c
where l defines the correlation length. The corresponding
c
spectraaregivenby
lc=0.1nm 0.3 1 3 10
q2 =q2g(kl ), (55)
h ik 1 c
where
g(x)=1 (1+x2) 2 (56)
− − Fig.1. Absorption spectrum due to a Disordered Charge
Distribution.Theabsorptioncoefficientα is giveninarbi-
and DCD
trary units, normalized at λ = 100µm and l = 10nm. The
c
ω
χ (ω)= q2 g(ω/ω ), (57) solid,dot,dash,dash-dotanddash-dot-dotlinescorrespondto
′0′ 12πυ3t m c lc =0.1,0.3,1,3and10nmrespectively,assumingatransverse
soundvelocityυ =3 105cm/s.
where t ·
ω =υ/l .
c t c of solids linked to disordering. Vacancies unavoidable in dis-
orderedlatticesproduceinfirstapproximationapossibilityfor
ThespectraldependenceofEq.(56)is
chaoticdistributionofsomecharges,whichwasconsideredin
ω ω2/ω2 forω ω the previoussection. This distribution is consideredas frozen
g ∼ c ≪ c (58)
(cid:18)ωc(cid:19)(≈1 forω≫ωc icnietshienlcaottmicpeadruiseontowthitehltahregeavhaeiliagbhlteotfhebramrraielrendeivrgidyi.nTghveaTcaLnS-
This is also true of other often used correlation function theorytakesintoaccounttransitionsofchargesduetoquantum
shapes,suchasstepcorrelationfunctions: tunnelingpossibleatanytemperature.
The TLS theoryis the result of a macroscopicanalysis of
δ(r) 3/(4πl3) for r <l ,
φ(r)= − c | | c (59) phenomenaasopposedtoanexactmicroscopicdescription.It
0 for r l .
( | |≥ c is based on the hypothesis of a flat distribution of tunneling
stateswithenergyleveldifferencessufficientlylowwithrespect
ThespectraldependenceinEq.(58)iscommonforcharge
totheusualvibrationalenergy.Thetemperaturedependanceof
distributions, which are uncorrelatedover short distances and
two-level system properties results from deviations from the
respectneutralityoverlargerscales.
model of harmonic oscillator, which is a good approximation
The DCD absorption is temperature independent. The
for the description of crystal lattice vibrations. The origin of
corresponding absorption coefficient α(ω) ωχ (Eq.15)
∼ ′o′ TLSisvacanciesin the lattice whichgivesthe possibility,for
presentstwoasymptoticsbehaviorsonbothsidesofω which
c
atomsorgroupsofatoms,tohavedifferentequilibriumspatial
dependsonthecorrelationlength.Inthehighfrequencyrange,
configurations. As a result, the potential curve has at least a
thespectralindextakesthevalueβ=2,andforthelowerfre-
double-wellform(DWP, see Fig. 2) in contrastto single-well
quency range, the spectral index is equal to β=4. This fre-
potential(SWP)ofharmonicoscillatorsincrystalwithoutva-
quencydependenceabsorptionis shownon Fig. 1 forvarious
cancies.
valuesofl .
c
IntheTLSmodeldescribede.g.by(Phillips1987),theen-
ergydistributionofstatesisdefinedundertheassumptionthat
4.3.TLSAbsorption theasymmetry∆ofDWPandtheheightV oftheDWPpoten-
tialbarrierarestochasticvaluesandtheirprobabilitydistribu-
4.3.1. PhenomenologicalTLSTheory
tioncan beassumedconstantovera certainrangeofV and∆
At low temperatures, the thermal and dielectric properties of values. Like forthe well knowntunnelingprobability,the en-
disordered solids (e.g. glasses and mixed crystals) show defi- ergy splitting ∆ has an exponential dependence with barrier
0
nitedeviationsfromthepredictionsofthephonontheorydevel- widthd:
opedforperfectlyorderedcrystals. Mostof these phenomena
can be describedin the formalismof the so called Two Level ∆ =~Ωexp[ d(2mV)1/2/~], (60)
0
−
Systems(TLS),whichisthesimplestmodeloftunnelingstates.
IncomparisontoDCD itcouldbeconsideredasasecondap- where ~Ω = (E + E )/2 is the average energy of levels and
1 2
proximation for the description of electromagnetic properties m is the effective mass. This exponential dependence along