theory applied to warm dense matter · 2017. 7. 11. · redmer, science 348, 1455 (2015) d...
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Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. SAND2017-6201 C
TheoryappliedtoWarmDenseMatterLukeShulenburger
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WhatisWarmDenseMatter?§ WarmDenseMatterisgenerallyassociatedwithstronglycoupledions
(Gii >1)andmoderatelydegenerateelectrons(q ~1)
§ Itistypicallyfoundatthejunctionofsolid,liquid,gas,andplasma.ThecomplicatedinterplayofthephysicalprocessesthatWDMshareswithitsneighborscreatesconsiderabledifficultiesfortheory.
§ q >>1isequivalenttowhereL isthethermaldeBrogliewavelength§ Fermi-Diracstatisticsforelectronicdegreesoffreedomstarttakingon
moreofaMaxwell-Boltzmanncharacter
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WhereisWarmDenseMatterfound?
is closely connected with (P > 1 Mbar)
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Somehistoryofthefield§ Conferencesdevotedtonon-idealplasmasandtheirstudy
havebeenaroundforalongtime§ ThePhysicsofNon-IdealPlasmasMeetingseriesstartedin1980informer
EastGermany§ TheStronglyCoupledCoulombSystemsmeetingshavealsohadalong-
standingcontingentinterestedinwhatisnowcalledWarmDenseMatter
§ Thefirst“WarmDenseMatter”meetingwasorganizedbyAndrewNg(whocoinedthename)andheldinVancouver,BCin2000.
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Whatmakesitdifficult?§ Thereisnosmallparameter
§ Plasmaexpansionsing,theplasmaparameter,failforWDM
§ Manydifferentaspectsofthephysicscontributeatacomparablelevelandmustbeincluded§ strongcorrelations§ ionization§ bondformationandbreaking§ complexpressureandtemperaturedependentchemistry
§ Computations/simulationsareoftenquitedemanding§ Massivelyparallelcomputationsarethenorm§ Shortcutsareverytempting
§ Experimentalconditionsareshortlivedandhardtodiagnose§ Whichexperimentalresultsshouldyoubelieve?
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Better EOSPhase diagrams
Time/Path dependence Approach
to phase equilibrium
Transport and mechanical properties
Mixtures
Material structure, Electronic, Ionic
Non-LTE
Electronic Excitation
Heating the Material
Ionic ExcitationCompressing the
Material
ScientificneedsinWarmDenseMatterResearch
Experimentalapproaches
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Considersomethingas“simple”asthedeuteriumHugoniot
7Pierre Henri Hugoniot
One of these thingsis not like the others
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Theprogresscontinuestothisday
§ Experimentaltechniquescontinuetoimproveinprecisionandaccuracy§ Understandingofmaterial
propertiesbootstrapsitself
§ Startingtodiscriminatethelimitationsofcurrenttheoreticalapproximations
8
Knudson and Desjarlais, PRL 118, 035501 (2017)
Pres
sure
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• Equations governing the properties of a material under any conditions are known
• Just need to solve the 3N dimensional partial differential equations• Approximations are necessary for real materials
QuantumCalculationsofferanappealingroute
9
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• Equations governing the properties of a material under any conditions are known
• Just need to solve the 3N dimensional partial differential equations• Approximations are necessary for real materials
• If we could do this accurately and efficiently, we could calculate any physical property
QuantumCalculationsofferanappealingroute
10
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Quantumcalculationsarenottrivial• RecastSchrodingerequationasanintegralproblemin3N
dimensions
● Massiveparallelismavailable,eachpointcanbecalculatedindependently
• Poorscaling
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Quantumcalculationsarenottrivial• RecastSchrodingerequationasanintegralproblemin3N
dimensions
● Massiveparallelismavailable,eachpointcanbecalculatedindependently
• Poorscaling� 3dimensionsperelectron� 20pointsineachdirection� 209 ≈512billionpointsfor3electrons
� 3.8TBjusttostore!
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DFT– TheMostCommonApproximation§ Threeinsightsunderpinthedevelopmentofthemost
commonlyusedtheory§ PhysicalInsight
§ Wavefunctionisnotanobservablebutthedensityis§ Replacethe3Ndimensionalwavefunctionwiththe3dimensionaldensity
§ Canapproximatekineticenergyanddevelopasensibledensitybysolvingfornoninteracting electronsinaneffectivepotential
§ Areasonableapproximationistomaketheeffectivepotentialasimplefunctionofthedensity
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N(r) = ∫Ψ(r,r2..rN)dr2���drN
V(r) ∝ 1/r
Ion
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DFT– TheMostCommonApproximation§ Threeinsightsunderpinthedevelopmentofthemost
commonlyusedtheory§ PhysicalInsight
§ Wavefunctionisnotanobservablebutthedensityis§ Replacethe3Ndimensionalwavefunctionwiththe3dimensionaldensity
§ Canapproximatekineticenergyanddevelopasensibledensitybysolvingfornoninteracting electronsinaneffectivepotential
§ Areasonableapproximationistomaketheeffectivepotentialasimplefunctionofthedensity§ Thisisthedensityfunctional
14
N(r) = ∫Ψ(r,r2..rN)dr2���drN
V(r) ∝ 1/r
Ion
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(Kohn-Sham)DFTinpractice§ Startbychoosingapproximation(functional)§ MakeBorn-Oppenheimerapproximation
§ Chooseachunkofmaterialtostudy(generallyperiodicboundaryconditions)
§ Solveforelectronsinpresenceofstaticions§ Note:Mermin approximationmeansthermalgroundstate
§ Hamiltonianisnowset.Solveforsingleparticlesolutionsinsomebasis(physiciststendtopreferplanewaves,chemistsgaussians)§ Note:allofthesesolutionsneedtobeorthogonalforFermionslike
electrons,soneedO(N3)diagonalizationofamatrix
§ Combinesingleparticlewavefunctionstogetdensityandthusinteractionpotential§ Usethistogetnewapproximationofpotentialanditeratetoself-
consistency 15
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WhatdoesDFTallowustocalculate?
§ Formally,wegetE(R)andrelatedquantitieslikeF(R)§ Ifweareintegratingovertheelectrons,thisisgenerally
enough§ Structure,Diffusion…
§ Forelectronicproperties,thingsaremorecomplicated§ Formallyspeaking,needanewtypeoffunctional§ Practically,weoftenuse(abuse)thesingleparticlewavefunctions
§ Conductivity,Opacity…
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DFTisaverysuccessfultechniqueforstudyingWDM
§ CarefulDFT/QMDcalculationscancomplementexperimentbyprovidingadditionalinformation
§ Thisisespeciallypowerfulwhenexperimentscanvalidateapproximations
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Root, LNS, Lemke, Dolan, Mattsson and Desjarlais, PRL 115, 198501 (2015)
Shock melting of diamond Phase diagram of MgO
Knudson, Desjarlais and Dolan, Science 322, 1823 (2008)
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DFTisnotperfect
§ Scalingwithtemperature§ Memory~T3
§ CPUtime~T4.5
§ Approximationsarenotoriouslydifficulttoimprove
§ Theseshortcomingsarenotjustacademic!
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Knudson, Desjarlais, Becker, Lemke, Cochrane, Savage, Bliss, Mattsson and
Redmer, Science 348, 1455 (2015)
D2 liquid-liquid phase transition
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FRONTIERSINCALCULATINGPROPERTIESOFWDM:EXTENDINGRANGEOFVALIDITY
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AverageAtomApproximation
§ Athightemperatures,thepresenceofnearbyatomsislessrelevant§ Solveforanisolated“averageatom”in
abackgroundofelectrons§ Selfconsistentlychangeionizationstate
withtemperature§ Fewerelectronson”atom”moreinbackground
§ InadditiontoEOS,ageneralizationtohandlevariousionicstatesinaplasmacanbeusedtocalculatetransportproperties
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Bailey et al. Nature 517, 56 (2015)
Fe opacity at stellar conditions
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AverageAtomApproximation
§ Athightemperatures,thepresenceofnearbyatomsislessrelevant§ Solveforanisolated“averageatom”in
abackgroundofelectrons§ Selfconsistentlychangeionizationstate
withtemperature§ Fewerelectronson”atom”moreinbackground
§ InadditiontoEOS,ageneralizationtohandlevariousionicstatesinaplasmacanbeusedtocalculatetransportproperties
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Bailey et al. Nature 517, 56 (2015)
Fe opacity at stellar conditions
What about higher densities / lower temperatures?See for example talk by C. Starrett
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OrbitalFreeDFT§ Kohn-Shamapproximationisnotonly
approachtoDFT§ Cansolveeverythingwithoutcalculating
singleparticlestates§ Approximationbecomesmuchmore
difficult§ Needtogetkineticenergyandelectronic
entropyfromdensityalone!§ Notoriouslydifficulttogetmoleculesto
bind§ However,thisisanoldtechnique
§ Thomas-Fermiapproximationisperhapsearliestexample
§ Typicallyworkswellathightemperatures
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SiO2 from OFDFT and KSDFT
Sjostrom and Crockett, PRB 92, 115104 (2015)
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OrbitalFreeDFT§ Kohn-Shamapproximationisnotonly
approachtoDFT§ Cansolveeverythingwithoutcalculating
singleparticlestates§ Approximationbecomesmuchmore
difficult§ Needtogetkineticenergyandelectronic
entropyfromdensityalone!§ Notoriouslydifficulttogetmoleculesto
bind§ However,thisisanoldtechnique
§ Thomas-Fermiapproximationisperhapsearliestexample
§ Typicallyworkswellathightemperatures
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SiO2 from OFDFT and KSDFT
Sjostrom and Crockett, PRB 92, 115104 (2015)
Work continues to improve functionals and implementationSee for example work of E. Carter, T. Sjostrom, S. Trickey and K. Burke
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PathIntegralMonteCarlo
§ ReturntodirectsolutionofmanybodySchrodingerequation
§ UseFeynmanformulationtocastasseriesofpathintegrals
§ SampleoverpathintegralsusingMonteCarlotechniques
§ Veryexpensive,potentiallyveryaccurate§ Difficultiesduetosymmetryof
particlesandergodicity(pathsgettangledatlowtemperatureorhighdensity)
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Hugoniot of Deuterium
Militzer and Ceperley, PRL, 85, 1890 (2000)
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PathIntegralMonteCarlo
§ ReturntodirectsolutionofmanybodySchrodingerequation
§ UseFeynmanformulationtocastasseriesofpathintegrals
§ SampleoverpathintegralsusingMonteCarlotechniques
§ Veryexpensive,potentiallyveryaccurate§ Difficultiesduetosymmetryof
particlesandergodicity(pathsgettangledatlowtemperatureorhighdensity)
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Hugoniot of Deuterium
Militzer and Ceperley, PRL, 85, 1890 (2000)
Work continues to improve approximations and expand validity for WDMSee for example work of Bonitz, Ceperely and talk by B. Militzer
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FRONTIERSINCALCULATINGPROPERTIESOFWDM:IMPROVINGAPPROXIMATIONS
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GroundstateMonteCarloApproaches§ Improvetreatmentof
electronicinteractionbystochasticsamplingofmanybodySchrodingerequation
§ Canbehighlyaccurate,butveryexpensive
§ Inaccuraciesareoftenduetoapproximationsnecessarytoimprovecomputationalcost§ Methodisverywellsuited
tosupercomputersthough,sothisshouldimprove
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Mazzola and Sorella, PRL 114, 105701 (2015)
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BetterDFTfunctionals
§ DFTgets10’softhousandsofcitationsyearly§ Lotsofworkonimproving
functionalsforgroundstate/ambientproperties
§ Functionalsforfinitetemperature§ OlderworkbyDharma-
Wardana andPerrot(1984)§ RecentPIMCcalculationsfor
referencesystemhavespurredwork
§ Trickey groupandBurkegroupbothhavenewapproximations
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Browne et al. PRL 110, 146405 (2013)
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FRONTIERSINCALCULATINGPROPERTIESOFWDM:BEYONDEQUATIONOFSTATE
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TransportProperties:Kubo-Greenwood
§ CancalculateelectronicandthermalconductivityusingDFT
§ CalculateimaginarypartofdielectricfunctionusingtheKubo-Greenwoodrelation§ Linearresponse§ Matrixelementsofsingle
particlewavefunctions
§ WorkswellinWDMregime§ Issuesifbandgapisnot
closed(Helium?)
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Desjarlais, Kress, and Collins, Phys. Rev. E 66, 025401 (2002)
Aluminum electrical conductivity
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Transportproperties:Energytransfer§ Stoppingafastmovingion§ Understandinghowionsareslowedisessentialtounderstanding
theenergybalanceininertialconfinementfusion§ Asatestproblem,wedragahydrogenionthroughaluminumat
constantvelocityandmeasuretheforceontheion§ Generationofplasmons necessarytocapturetheproperbehavior
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Born-Oppenheimer TDDFT
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Transportproperties:Energytransfer§ Stoppingafastmovingion§ Understandinghowionsareslowedisessentialtounderstanding
theenergybalanceininertialconfinementfusion§ Asatestproblem,wedragahydrogenionthroughaluminumat
constantvelocityandmeasuretheforceontheion§ Generationofplasmons necessarytocapturetheproperbehavior
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Born-Oppenheimer TDDFT
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Transportproperties:Energytransfer§ Stoppingafastmovingion§ Understandinghowionsareslowedisessentialtounderstanding
theenergybalanceininertialconfinementfusion§ Asatestproblem,wedragahydrogenionthroughaluminumat
constantvelocityandmeasuretheforceontheion§ Generationofplasmons necessarytocapturetheproperbehavior
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Born-Oppenheimer TDDFT
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FRONTIERSINCALCULATINGPROPERTIESOFWDM:WHERETOFROMHERE
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Whatproblemsseemripe?
§ WhiteDwarfs§ Cosmochronology – ageofwhitedwarfsthroughcoolingrate
§ EquationsofstateforC,O,andHeatextremeconditions§ Thermalconductivities,viscosities,diffusioncoefficients
§ Asteroseismology – modelpulsatingwhitedwarf,toinfer§ Totalmassandmasscomposition§ Interiorrotationprofile§ Surfacetemperature§ Structuraldetails
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SummaryComments
§ WarmDenseMatterresearchisgrowingrapidlyandaddressingimportantproblemsininertialfusion,planetaryscience,andmanyotherfields
§ WarmDenseMatterhasbeenveryslowtoyieldtopurelytheoreticaldescription
§ Advancesinelectronicstructureandmoleculardynamicsmethodshaveplayedanenormousroleinadvancingourunderstandingofwarmdensematter
§ Thisisgoingtobeaveryinterestingweek!36