Study of interatomic potentials in ZnS - Crystal-GRID experiments versus ab initio calculations

7 Pages·0.47 MB·English

by  Koch, T.

Volume105,Number1,January–February2000
Journal of Research of the National Institute of Standards and Technology
[J.Res.Natl.Inst.Stand.Technol.105,81(2000)]
Study of Interatomic Potentials in ZnS—
Crystal-GRID Experiments Versus
Ab Initio Calculations
Volume 105 Number 1 January–February 2000
Timo Koch Crystal-GRIDmeasurementshavebeenper- Keywords: atomiccollisions;Crystal-
formedwithZnSsinglecrystals.Forthe GRID;gammarayspectroscopy;inter-
Forschungszentrum Rossendorf, firsttime,anasymmetricCrystal-GRIDline atomicpotentials;MolecularDynamics
D-01314 Dresden, Germany shapecouldbeobserved.Theprelimi- simulations;nuclearlifetimes;ZnS.
and narydataevaluationindicatesthatthere-
portedlifetimeofthe3221keVlevelin
Institut Laue-Langevin, F-38000
33Sistooshort.Avalueofabout60fshas Accepted: July22,1999
Grenoble, France beenfound.Duetothis“long”lifetime
thelineshapeismuchlessstructuredthan
Karl-Heinz Heinig calculatedwiththereportedlifetime. Availableonline: http://www.nist.gov/jres
Forschungszentrum Rossendorf,
D-01314 Dresden, Germany
Michael Jentschel
Institut Laue-Langevin, F-38000
Grenoble, France
and
Hans G. Bo¨rner
Institut Laue-Langevin, F-38000
Grenoble, France
1. Introduction
During the last decade the Gamma Ray Induced GRID”, should facilitate the study of interatomic solid
Doppler broadening (GRID) technique has been opti- statepotentials,lifetimesofnuclearstates,andimpurity
mised at the Institut Laue-Langevin (ILL). It has been locationsinthelattice.Inrecentworks[3,4,5]thegen-
successfully applied to lifetime measurements. A de- eral applicability of the new method to the study of
scription of the method can be found in Ref. [1]. interatomic potentials has been proven.
Theuseofsingle-crystallinesamplesforGRIDexper- In this paper a preliminary report on measurements
imentswasproposedduringthefirstGRIDworkshopin withZnSsinglecrystalswillbepresented.Theseexper-
1992 [2]. This special application, called “Crystal- imentsaimedtoobserveforthefirsttimeatheoretically
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Journal of Research of the National Institute of Standards and Technology
predictedasymmetry[2]oftheDopplerbroadenedline it represents a sensitive indicator for the interatomic
shapeforthe(cid:1)111(cid:2)crystaldirectionalignedtothespec- potential.
trometer axis. It is expected that such an asymmetry Crystal-GRIDmeasurementscanbeperformedatthe
allows the determination of interatomic solid state po- high-fluxneutronreactoroftheInstitutLaue-Langevin
tentials and nuclear lifetimes to a higher precision than inGrenoble.Thecrystalstobeinvestigatedareplacedin
with symmetric line shapes. adistanceof55cmfromthereactorcoreinatangential
beam tube, where they are permanently irradiated by
thermal neutrons (≈5(cid:2)1014 neutrons/cm2s). In gen-
2. The Crystal-GRID Technique eral, three oriented crystals of size 17 mm(cid:2)20
mm(cid:2)2mmareusedforthemeasurements.Twohigh-
In a thermal neutron capture reaction the binding resolution gamma spectrometers, GAMS 4 and GAMS
energyoftheneutronisdepositedinthenucleusandthe 5, are placed on either side of the beam tube at a dis-
deexcitation takes place via a (cid:1)-cascade. The emission tance of 13 m and 17 m of the samples, respectively.
of a first, primary photon (cid:1) entails a recoil of the still Bothoperateindoubleflatcrystalgeometry,i.e.energy
1
excitednucleusduetotheconservationofmomentump: resolution is gained by successive Bragg reflection on
two almost perfect single crystals.
p=mv =E /c, (1)
0 (cid:1)1
where m is the mass and v the initial velocity of the 3. RequirementsofCrystalsToBeStudied
0
recoilingnucleus,E theenergyoftheprimaryphoton
(cid:1)
1
and c the velocity of light. As the energy of the first Theoretically, Crystal-GRID studies can be per-
photonisoftheorderofseveralMeV,theinitialvelocity formed with a large number of materials, however the
v of the recoiling atom lies in between 10–5 c and sensitivity of the Doppler-broadened line shapes to the
0
2(cid:2)10–4 c, corresponding to an initial kinetic energy atomicinteractiondependsonanumberofcriteria.First
E ofapproximately100eVto1000eV.Thedirection ofall,thematerialshouldcontainanisotope,whichhas
kin,0
oftheinitialrecoilisarbitrary.Thefurthermotionofthe a simple (cid:1) cascade with a mostly primarily fed short
recoiling nucleus is defined by the interatomic forces, living(5fsto40fs)level.Theprimaryfeedingguaran-
i.e., by the interatomic potential in the solid. tees an almost monoenergetic recoil energy, the short
At the time t' a second photon (cid:1) is emitted and lifetimetherestrictiontoafewcollisionprocesses.Fur-
2
observedbythespectrometerifemittedalongthedirec- ther, the coordination number of the material itself
tionofobservation.Ifthenucleus[velocityv(t)]hasnot shouldbeaslowaspossible.Inthiscasetheanisotropy
yet come to rest, the (cid:1) energy is Doppler shifted by of the recoil motion and therefore the fine structure in
the Crystal-GRID line shape is strongly pronounced.
v(t')(cid:4)n Inanexperimentanumberofadditionalrequirements
(cid:3)E =E0 , (2)
(cid:1)2 (cid:1)2 c have to be met: Currently, the intensity requirement is
themostlimiting.Forthatreasonsufficientlybigsingle
whereE0 istheunshiftedenergyofthesecondphoton crystals are needed. The typical size used up to now is
(cid:1)
2
andntheunitvectorinthedirectionofobservation.Due three pieces 17 mm(cid:2)20 mm(cid:2)2 mm. As soon as
to the total number of Doppler shifted photons a GAMS5isoperationalinbent-crystalmode,thinsingle-
Dopplerbroadenedphotonenergyspectrumisobtained. crystallinefilmsshouldalsobeusable.Further,themea-
The motion of the recoiling atoms within a single sured intensity depends on a number of quantities such
crystal is highly anisotropic. For example, the slowing as the (cid:1) capture cross section, the absolute intensity of
down in the direction of the nearest neighbour the (cid:1) transition, and the response function of the spec-
(“blocking”) is more efficient than in the direction be- trometer, which itself depends on the (cid:1) energy as well
tween two nearest neighbours (“channeling”). There- asthethicknessandthematerialofthecrystalsusedin
fore,theisotropicdistributionofvelocitiesjustafterthe the spectrometer. According to the requirement of a
recoil becomes anisotropic as soon as the interaction simple (cid:1)-cascade with short lived levels, light nuclei
with nearest neighbours becomes non-negligible. In a would be preferential. However these isotopes do often
powder target this anisotropy is averaged out by the have a very small neutron capture cross section. This
arbitrary alignment of the micro-crystals. However, in limits currently the number of candidates for Crystal-
thecaseofasinglecrystalthisanisotropyexistsmacro- GRID experiments to isotopes such as 36Cl, 49Ti, 56Fe,
scopically and leads to a fine structure in the Doppler 59Ni,52Cr,and33S.Thereforeimprovementsofthespec-
broadenedlineshape.Asthefinestructureiscausedby trometerefficiencyareurgentlyneeded.Afirststepwill
theinteractionoftherecoilingatomwithitsneighbours be done with GAMS 5. An efficiency improved by a
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Journal of Research of the National Institute of Standards and Technology
factor of 100 would be needed, in order to be able to (RMD) and is sufficient for Crystal-GRID simulations
investigate alot of materials that are of great interest in [3].
solidstatephysics,forexample,anumberofimportant For ZnS 1400 trajectories r(t) with velocities v(t)
i i
semiconductors. havebeensimulatedwithanRMDprogram.Forbetter
Becausethecrystaltemperaturecanreach1200Kin statistics the symmetry operations of the appropriate
thereactor,thecrystalstructureneedstobestableupto pointgroupcanbeappliedtothetrajectories.Inorderto
this temperature region. Furthermore reactor safety re- compare simulation and experiment the set of trajecto-
quirements must be considered ((cid:5)-activity, no melting riesneedstobetransformedtoaprobabilitydistribution
or evaporation of material, etc.). of velocities or energies. In an actual measurement the
Oncethesebasicrequirementsarefulfilled,theoreti- directionofobservationisfixed.AsseeninEq.(2),the
calcalculationscanbeperformed(seeSec.4).Ifaline Dopplershiftonlydependsontheprojectionv =v(cid:4)nof
(cid:3)
shape is very structured or not, depends mainly on the the velocity on the direction of observation.
lifetimeandthecrystallattice.Shortlifetimesandopen For a given n the velocity projection distribution
structures lead to the most pronounced line shapes. P(v,t) dv dt can be deduced by counting the trajecto-
(cid:3) (cid:3)
ries having a velocity projection v(t)(cid:4)n in the range
i
v...v +dv duringthetimeintervalt...t+dt.Anormal-
(cid:3) (cid:3) (cid:3)
4. Theoretical Description isation of the distribution is not necessary, because the
measurement does not yield absolute intensities. This
Ananalyticalcalculationoftheoreticallineshapesis distribution is independent of the lifetime (cid:6)of the nu-
only possible for simple assumptions on the slowing clearstate,becausethetrajectorieshavebeencalculated
down process. Crystal-GRID line shapes can not be for a fixed period of time regardless of the second (cid:1)
deducedwithsufficientaccuracyinthatway.Wehaveto decay.
use computer simulations instead. As the recoil veloc- ThedistributionP(v)dv ofvelocityprojectionsv at
(cid:3) (cid:3) (cid:3)
itiesoftheatomsaremuchbelowtypicalelectronveloc- thetimeofthesecond(cid:1)decaycanthenbecalculatedfor
ities(≈7(cid:2)10–3c),electronicexcitationsarenottaken a given lifetime (cid:6)by multiplying the radioactive decay
into account. This allows consideration of the recoil law:
motion within classical mechanics. The most exact (cid:4) (cid:5) (cid:6)
computersimulationapproachwithinclassicalmechan- 5(cid:6) t'
P(v) dv = exp (cid:7) P(v,t') dt' (3)
icsaremoleculardynamics(MD)calculations.Inthese (cid:3) (cid:3) (cid:6) (cid:3)
0
calculations the motion of N atoms within a finite vol-
ume V is calculated assuming certain initial and Theintegraliscalculatedupto5(cid:6),where99.3%ofthe
boundary conditions and the knowledge of the atomic secondaryphotonshavebeenemitted.UsingEq.(2)this
interactionofallNatoms.Thetrajectoriesoftheparti- canbetransformedtotheprobabilitydistributionP(E )
(cid:1)
2
cles can be calculated by simultaneously integrating dE oftheDopplershiftedenergy,thesocalledCrystal-
(cid:1)
2
Newton’s equations of motion of all N particles. In the GRID line shape
caseofrecoilsimulationsforCrystal-GRIDexperiments (cid:5) (cid:6)
theinitialconditionsaredefinedbythecrystalstructure c(cid:3)E
ofthetargets,whereallatomsmoveataninitialvelocity P(E(cid:1)2) dE(cid:1)2(cid:7)P v(cid:3)= E(cid:1)0(cid:1)2 dE(cid:1)2 (4)
2
following a Maxwell distribution corresponding to the
temperature of the crystal. The simulation volume is a Thislineshapewouldbemeasuredwithaspectrom-
cubic cell, the size of which is adjusted such that the eteryieldinga(cid:8)-likeresponsefunction.Inthecaseofan
recoiling atoms do not reach the boundaries within the ideal double flat crystal spectrometer the instrumental
time of simulation. Periodic boundary conditions are response function can be calculated from diffraction
used.Startingwithaninitialrecoil,possibletrajectories theory. Due to small imperfections of the used spec-
can be simulated. The method is exact within classical trometer crystals an additional broadening has to be
mechanics. Hence the result of MD simulations is as taken into account by folding the ideal instrument re-
goodastheinteratomicpotential.However,MDsimula- sponse function with a Gaussian of width (cid:9) , the so
EW
tionsareverytimeconsuming.Forourpurposeonlythe called “excess width”. The later can be determined ex-
interaction of the recoiling nucleus with all its neigh- perimentally. When folding P(E ) with the obtained
(cid:1)
2
bouring nuclei needs to be known. The interaction of realisticresponsefunction,thelineshapetobefittedto
two nuclei far away from the recoiling nucleus has no theexperimentaldatacanbecalculated(fordetailssee
appreciable influence and can be neglected. This ap- [1]).
proximation is called “restricted molecular dynamics”
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Journal of Research of the National Institute of Standards and Technology
As can be seen, the measured Crystal-GRID line of the crystals. Crystals having an inversion symmetry
shapesaredirectlycorrelatedtotheslowingdownpro- yieldsymmetriclineshapesforallorientations,because
cess in the crystal. For a given crystal structure and thenucleusslowsdowninthesamewayforanemission
alignment of the crystal, the calculated line shape de- withvelocity+v or(cid:7)v.TheGRIDmethodisinsensi-
0 0
pends only on the classical potential used in the RMD tive to rotations of the sample around the direction of
calculationandonthelevellifetime.Therefore,bycom- observation. Therefore a symmetric line shape is also
paringsimulationandmeasurement,theinteratomicpo- producedifthelatticeonlyhasarotation-inversionsym-
tential may be improved in the energy region of 10 eV metry around this axis.
to 1000 eV. Thezincblendestructurehasnoinversionsymmetry.
However a rotation-inversion symmetry exists for some
axes,e.g.,forthe(cid:1)100(cid:2)andthe(cid:1)110(cid:2)directions,butnot
5. The Zincblende Structure for the (cid:1)111(cid:2) direction. Consequently the expected line
shape in the latter case is asymmetric.
Undernormalconditions,Zincsulfide(ZnS)hasthe
zincblendestructure(pointgroupF43m).Thetwocon-
stituentsarefoundontwoface-centredcubicsublattices 6. Experiments
respectively,displacedwithrespecttoeachotherbyone
forth of the diagonal of the cubic unit cell. In GRID DuringtwomeasuringperiodsweexaminedtheZnS
experimentsthedirectionofobservation,i.e.,thedirec- crystalswiththe(cid:1)111(cid:2)and(cid:1)110(cid:2)directionsrespectively
tionoftheaxisbetweenthesampleandthespectrome- oriented towards the spectrometer. The measurements
ter, is well defined. usedthe2380keVtransitioninthe33Snucleus,depop-
InpreviousCrystal-GRIDmeasurementstheDoppler ulatingthe3221keVlevel.Thislevelismostlydirectly
broadeningofthelineshapewasalwayssymmetricwith fed, 91% of the populating nuclei decay directly from
respect to the unshifted photon energy. This symmetry the capture state (see Fig. 1). Two two-step cascades
wasduetotheexistinginversionsymmetryinrealspace furthercontributetothefeeding[6].Theirinfluenceis
Fig.1. Partiallevelschemeof33S[6].ForeachlevelofenergyE thereportedlifetime
level
(cid:6)isgiven[7].ForthetransitionsthephotonenergyE(cid:1),thecorrespondinginitialrecoil
velocityv oftheSnucleus,andtheabsoluteintensityofthetransition(%meansper100
0
neutrons)aregiven.
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Journal of Research of the National Institute of Standards and Technology
small but taken into account in the RMD simulations. Duetothelonglifetimethelineshapesaremuchless
Theenergyofthefirstphotonis5421keV,correspond- structured than predicted by the computer simulations
ingtoaninitialrecoilenergyoftheexcitedSnucleusof performedbeforetheexperimentwiththereportedlife-
478eV,i.e.,arecoilvelocityof1.76(cid:4)10–4cor0.529Å/fs. time.Thus,theinformationwhichcanbegainedforthe
Thenuclearstatehasareportedlifetimeof(40(cid:10)12)fs interatomic potentials will be reduced.
[7].
The double crystal spectrometer was used with one
crystal in first and the other in second reflection order. 7. Ab Initio Results
For each orientation of the crystal 35 scans have been
measured.Everyscanconsistedof90datapoints,each Our second, purely theoretical approach to ZnS po-
of them being measured for 320 s. Thereby a peak tentialsconsistsofquantummechanical—socalledab
intensityofabout40countsperscancouldbeattained. initio—calculations.Thesecalculationsallowtoobtain
Thedataevaluationisstillinprogress.Uptonowwe the total energy of an arbitrary configuration of typi-
obtained two preliminary results. On the one hand, for cally64atomswithoutrequiringanyinformationabout
thefirsttime,thepredictedasymmetryofthelineshape classical interatomic potentials. By calculating the en-
in the (cid:1)111(cid:2) direction could be confirmed (see Fig. 2). ergyofaseriesofatomicconfigurations,aninteratomic
Ontheotherhandthelifetimeofthenuclearstateunder potential can be developed theoretically.
investigation was found to be much longer than the re- InafirststeponeoftheSatomswithinasimulation
portedvalue[7].Consideringtwodifferentinteratomic cellof64atomswasdisplacedalongthe(cid:1)111(cid:2)direction
potentials (ZBL [8], KrC [9]) in our theoretical model towardsitsnearestneighbour.Fordifferentdistancesthe
and fitting the results of the two crystal orientations total energy has been calculated. The difference to the
independently,weobtainalifetimearound60fsascan totalenergyoftheideallatticegivesafirstapproachto
beseeninFig.3.Thefittedlifetimesforthetwodiffer- anabinitiopotential(seeFig.4).Eventhoughnotonly
ent directions are not identical. This indicates that the the Zn—S interaction between two particles is consid-
interatomicpotentialsusedinthesimulationdonotwell ered, the contribution of other atom pairs should be
describe the interaction. Consequently a new potential comparatively small.
will be developed.
Fig.2. Doppler-broadenedlineshapeofthe2380keVtransitionin33Sforthe(cid:1)111(cid:2)direction.Thelinerepresents
thefittedtheoreticallineshapeobtainedwithRMDsimulationsusingtheZBLpotentialandalevellifetimeof
65fs.Theasymmetrycanbeseeninthewingsofthecurvearound120eVandinthesmallpeaksbetween200
eVand300eV.
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Journal of Research of the National Institute of Standards and Technology
RMDcalculationshavebeenperformedwiththisde-
duced ab initio potential. As can be seen in Fig. 3 the
obtainedlifetimeisabout70fs.Theminimum(cid:11)2value
of the fit, however, is higher, i.e., worse, than for the
screenedCoulombpotentials.Thisindicatesthattheab
initiopotentialdoesnotwelldescribetheslowingdown
inthecrystal.Thismaybeduetosomecrucialapprox-
imations within the ab initio calculations, e.g., with
respect to the exchange energy, that have to be made.
Furtherapproacheswillbemadeinordertoimprovethe
potential.
8. Conclusion
Crystal-GRID measurements have been performed
withZnSsinglecrystals.Forthefirsttime,thetheoret-
icallypredictedasymmetriclineshapecouldbeverified
experimentally. At the actual state of data evaluation it
seemstobeclearthatthelifetimeofthe3221keVlevel
in33Sisabout60fs.Thisvalueiswellabovetheprevi-
ouslyreportedvalue.Thereforethelineshapesareless
structured than predicted.
Duetothereducedinformationthatcanbeextracted
fromthelineshapes,athirdZnSmeasurementhasbeen
Fig. 3. Determination of the lifetime of the 3221 keV level in 33S. proposed. It should be performed in the (cid:1)100(cid:2) orienta-
Computer simulations have been performed for different screened
tion.Withtheadditionalresultsitshouldbepossibleto
Coulombpotentials(KrC,ZBL).Foreachcrystalorientationtheoret-
validate interatomic potentials even though, due to the
icallineshapesfordifferentlifetimescanbefittedtotheexperimental
data. The calculated (cid:11)2 values are plotted against the lifetime. The “long”lifetime,thetransitionisnotoptimalforCrystal-
minimaofthecurvesgivetheresultinglifetimes.Theyvaryaround60 GRID studies.
fs.Usingthepreliminarypotentialobtainedfromabinitiocalcula-
tions, a lifetime around 70 fs would be obtained. However the (cid:11)2
values are worse than for the screened Coulomb potentials. The ab
initioresultsneedtobeevaluatedmorecarefully.
Fig. 4. The energy of a 64-atom lattice with one S atom displaced along the (cid:1)111(cid:2)
directiontowardsthenearestneighbourhasbeenabinitiocalculated.Thedifferenceto
theenergyoftheideallatticeisplottedversustheZn—Sdistance.Thenearestneighbour
distanceintheideallatticeis2.33Å.Theobtainedrelationcanbeseenasanabinitio
potentialandiscomparedtotheclassicalscreenedCoulombpotentialsusedinthiswork
(KrC,ZBL).
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Volume105,Number1,January–February2000
Journal of Research of the National Institute of Standards and Technology
Thefirstabinitioresultsdonotyetyieldsufficiently
goodresults.Neverthelessthesimultaneousexperimen-
talandtheoreticalapproachesallowadirectcomparison
of the results, leading to an increased reliability of the
obtained potentials.
9. References
[1] H.G.Bo¨rnerandJ.Jolie,Sub-picosecondlifetimemeasurements
by gamma ray induced Doppler broadening, J. Phys. G: Nucl.
Part.Phys.19,217–248(1993).
[2] K. H. Heinig and D. Janssen, The fine structure of Doppler
profilesinmoleculardynamicschannellingcalculation,ILLIn-
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[3] M.Jentschel,Crystal-GRID:EineneuenukleareSondezurUn-
tersuchung atomarer Bewegung im Festko¨rper. PhD thesis, TU
Dresden(1997).
[4] M. Jentschel, K. H. Heinig, H. G. Bo¨rner, J. Jolie, and E. G.
Kessler, Atomic collision cascades studied with the Crystal-
GRIDmethod,Nucl.Ins.Meth.B115,446–451(1996).
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GRID-techniqueinnuclearandsolidstatephysics,inProceed-
ingsofthe9thInternationalSymposiumonCaptureGamma-Ray
Spectroscopy and related topics (CGS 9, Budapest, Hungary,
8–12October1996),G.L.Molna´r,T.Belgya,andZ.Re´vay,eds.,
Springer(1996)pp.755–768.
[6] S.Raman,R.F.Carlton,J.C.Wells,E.T.Jurney,andJ.E.Lynn,
Thermalneutroncapturegammaraysfromsulfurisotopes:Ex-
perimentandtheory,Phys.Rev.C32,18–69(1985).
[7] R. B. Firestone, Table of Isotopes. New York, Wiley-Inter-
science,8thed.(1996).
[8] J.F.Ziegler,J.P.Biersack,andU.Littmark,TheStoppingand
RangeofIonsinSolids,Vol.1ofTheStoppingandRangesof
IonsinMatter.NewYork,PergamonPress(1985).
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Abouttheauthors: TimoHauschild(TimoKochatthe
time of the meeting) is a thesis student at ILL/FZ
Rossendorfwithspecialinterestsininter-atomicpoten-
tials. Dr. Karl-Heinz Heinig is a Staff Scientist at FZ
Rossendorf, Institute of Ion Beam Physics and Materi-
alsResearch.Hisspecialinterestsincludemodelingof
ionimplantationprocessesandnanoclusterformation.
Dr.MichaelJentschelisaPostDoctoralfellowatILL.
His special interests are high resolution gamma spec-
troscopy and atomic collision processes. Dr. Hans
Bo¨rner is the Group Leader for Nuclear and Particle
PhysicsatILLwithspecialinterestsinhighresolution
gamma spectroscopy, nuclear structure at low excita-
tion energies, and fundamental physics with neutrons.
87