Polarizability measurements in a molecule near-field interferometer

0.24 MB·English

by  Martin Berninger

APS/123-QED
Polarizability measurements in a molecule near-field interferometer
∗
Martin Berninger, Andr´e Stefanov, Sarayut Deachapunya, and Markus Arndt
Fakult¨at fu¨r Physik der Universit¨at Wien,
Boltzmanngasse 5, A–1090 Wien
(Dated: February 9, 2008)
Weapplynear-fieldmatter-waveinterferometry todeterminetheabsolutescalar polarizability of
thefullerenesC60 andC70. Akeyfeatureofourexperimentisthecombinationofgoodtransmission
and high spatial resolution, gained by wide molecular beams passing through sub-micron gratings.
Thisallowstosignificantlyfacilitatetheobservationoffield-dependentbeamshifts. Wethusmeasure
thepolarizability to be α=88.9±0.9±5.1˚A3 for C60 and to α=108.5±2.0±6.2˚A3 for C70.
7 PACSnumbers: 03.65.-w;03.75.-b;06.20.-f;33.15.Kr;39.20.+q
0
0
2 Knowing the scalar polarizability of a molecule is of termined by measuring the lateral shift of the molecular
importanceinvariousareasofphysics. Thepolarizability beam due to this field. In most experiments the result-
n
mayprovideone ofseveralparametersinthe description ing deflections are much smaller than the width of the
a
J of molecular shapes [1]. Even for the structurally simple molecularbeamprofile [2] butstill suchexperiments can
molecules C and C , various theoretical models have typicallyreachanaccuracyofabout10% [14]. Onemay
5 60 70
2 been competing with each other [2]. also consider the use of electric field gradients to induce
Themolecularresponsetoexternalelectricfieldsisalso a longitudinal velocity shift and corresponding time-of-

1 the handle for slowing [3, 4, 5] cooling or trapping [6] flight delay in a molecular fountain arrangement[15].
v
neutralparticlesinswitchedelectricfieldsoroff-resonant For atoms, very precise polarizability values were also
9
laser beams. obtained using far-field matter-wave interferometry: the
7
1 Italsoturnsouttobecrucialformatter-waveinterfer- atomic wavefunction can be coherently split and guided
1 ometry. The van der Waals or Casimir-Polderpotentials through two spatially separated electric fields. The dif-
0 acting between the traversing molecules and the diffrac- ferentphasesimprintedontheatomthroughtheinterac-
7 tion gratings have an enormous influence on the details tionbetweenαandelectricfieldE alongthetwodifferent
0 of the interference fringes [7, 8]. And this will generally pathsshowsupasafringeshiftthatcanbewellresolved.
/
h grow with the the size of the particles. This allowed a very accurate determination of ground
p Also the small-angle scattering cross section between state polarizabilities for Sodium [16] and Lithium [17].
- different molecules is related to their polarizability. And Efficient methods are also highly desirable for metrol-
t
n this enters, for instance, experimental considerations ogy on large molecules, in particular since their α/m-
a through the definition of the vacuum conditions which values may be significantly smaller than those of alkali
u
arerequiredfor avoidingcollisionaldecoherenceininter- atoms. The use of far-field interferometry would, how-
q
: ferometry [9]. ever,requirebeamsourcesofveryhighspatialcoherence.
v Finally,preciseα-measurementsarealsostimulatedby In the absence of efficient cooling and phase-space com-
i
X the insight that future molecule interferometers will also pression methods, this can only be achieved by strictly
r exploit optical gratings [10, 11, 12]. There the polariz- collimating the molecular beam. We have demonstrated
a ability is needed to determine the phase shift caused by far-field diffraction successfully in our earlier work for
the interaction between the induced dipole moment and large carbon molecules [18] but the limited flux of al-
the electric field of the diffracting laser beam. mostallotherlargeparticleswouldprohibitsuchastrict
Variousexperimentalapproachesareconceivabletode- spatialselection. While classicalStark deflection usually
termineαwithhighprecision[2]. Insolidsorliquids,the provides a high flux with a low spatial resolution, far-
polarizabilitycanbe obtainedthroughameasurementof field interferometers trade their high-resolution in for a
thedielectricconstantandtheindexofrefraction[1]. But strongly restricted molecular transmission.
in particular for the comparison with theoretical models Here we report on a solution of this problem, which
it is desirable to get information about the isolated par- isbasedonanear-fieldTalbot-Lauinterferometer(TLI).
ticles [13]. It combines high spatial resolution with high molecular
Polarizabilities of dilute molecular beams are usually throughput and in can be generalized to objects of arbi-
characterized in Stark-deflection experiments, in which trarymass—inthelimitofclassicalMoir´edeflectometry.
the neutralmoleculesaretraversingahomogeneouselec- A Talbot-Lau interferometer uses near-field wave ef-
trostatic force field. The polarizability can then be de- fects,generallyforthelenslessimagingofperiodicnanos-
tructures[19]. Ithasalreadybeendescribedearlierinthe
contextofatom[20,21]andmolecule[8,22]interferome-
try. InourpresentapparatusweemployaTLIcomposed
∗Electronicaddress: [email protected] ofthree goldgratingswith a gratingperiodofg=991nm
2
with a deflection voltage of 0kV and 6kV. The field-
lan dependent shifts can be nicely resolved and measured
gis Ds with high accuracy. In order to obtain additional statis-
scan
Laser beam ticalinformationandtoassesstheaccuracyofthevarious
Mask contributingparameters,werepeatthemeasurementsfor
Channeltron
z Diffraction Grating Detector different velocity distributions and for different deflec-
y Grating
tion voltages. We employ a gravitational velocity selec-
x Beam profile
Deflector scan
Source
Grating
Ds
800
Height
d
Delimiter
Molecular s 700
Beam L 0
1
r
e
p 600
FIG. 1: Experimental set-up: The three gratings are used nts
for coherence preparation, diffraction and detection. Path- u 500
o
dependentmatter-wavephaseshiftsin theexternalfield lead C
toabeamdeflectionatthepositionofthemaskgrating. The 400
lateral shift ∆s of the interference fringes at the detector is
directlyproportionaltothescalar molecularpolarizability α.
300
52.0 52.5 53.0 53.5 54.0
rd
Position of the 3 grating ((cid:181)m)
and slit openings of about 450nm width. FIG. 2: Deflection of a C60 beam with v¯ = 117m/s and
The first grating prepares the required spatial coher- σv = 8%. A phase shift of ∆φ = π is obtained at a voltage
encetoobtainmatterwavediffractionatthesecondgrat- of6kV(fullcircles). Theopencircles representthereference
ing,whichthenresultsinaregularmoleculardensitypat- at U=0kV.
tern, immediately in front of the mask grating. The in-
terference fringes, whose periodicity equals the gratings
constant, can be revealed by scanning the third grat- tion [8, 23] and chose the velocity by moving the source
ing laterally. The resulting modulation of the molecular to its appropriate vertical position. An interference pat-
flux behind this setup allows to visualize the molecular tern is then recorded by displacing the third grating in
nano-fringesystem. The molecularbeam width in a TL- steps of20nm overaboutthreefringe periods andby re-
interferometer may be wider than 1mm, allowing a high peating this scan for allvoltages within 3–15kV in steps
throughputofmoleculeswhiletheresolutionofthefringe of 1kV.
system can be better than 15nm.
To monitor and numerically compensate for drifts, an
We now insert an electrostatic deflector into the TLI
additional reference point (with U=0kV) is always in-
(seeFig.1),whichcreatesahomogeneousforcefieldF=
cludedbeforeandaftereachhigh-voltagedeflectionscan.
α(E∇E) changing the momentum of the molecules in
From the interference curves thus obtained we extract
proportion to the applied electric field E, its gradient
the voltage dependence of the experimental fringe shift
∇E and to the polarizability α. In the proper quantum
(Fig.3). Theobservedfringeshiftisinfluencedbythede-
picture, the electric field adds a path-dependent phase
tails ofthe molecular velocitydistributionin three ways.
shiftonthemolecularmatterwave. Inbothdescriptions,
Firstly, slow molecules will acquire a larger deflection in
classical and quantum, we obtain a fringe shift:
the external field gradient than fast ones (Eq. 1). Sec-
(E∇)E d d ondly,differentvelocityclassesareassociatedwithdiffer-
x
∆s (α,v)=α +L (1) ent visibilities. This is an important and desired feature
x m v2(cid:18)2 (cid:19)
of the Talbot-Lau arrangement,as it allows to prove the
quantum wave-nature of material objects [8]. Thirdly,
where m corresponds to the molecular mass, v the ve-
thevanderWaalsinteractionwiththegratingwallsadds
locity along the direction of the beam and L the dis-
a dispersive phase shift [7, 8].
tance between the source grating and the deflector. The
front edge of the deflector (length d) is positioned at Theexpectedsignalasafunctionofthescanninggrat-
L = (26.6±0.1)cm behind the first grating. The de- ing position x is proportional to:
flector provides a constant force with deviations of 0.5%
alongthex-axisovertheentiremolecularbeamdiameter.
2π
Two interference recordings at two different deflection 1+V¯ (α)cos x−∆s (α)
th th
voltages would be sufficient to determine the molecular g
(cid:2) (cid:0) (cid:1)(cid:3)
polarizability, if all parameters were known with arbi- 2π
≡ dvf (v) 1+V(v)cos x−∆s (α,v) (2)
trary precision. Figure 2 shows a fringe system of C60 Z v¯,σv (cid:18) g x (cid:19)
(cid:2) (cid:0) (cid:1)(cid:3)
3
collisions–thanthevisibility,andthevaluesgivenbelow
3500 are therefore based on the displacement measurements.
We find a polarizability of 88.9±0.9±5.1˚A3 for C
60
3000 and108.5±2.0±6.2˚A3 for C , including the statistical
70
(first) and systematic (second) error. The relative po-
2500
m) larizability is then α(C70)/α(C60) = 1.22±0.03. Most
n 2000 of the systematic uncertainties, such as for instance field
Shift ( 1500 irnehlaotmivoegveanleuietsie.s, cancel out in the determination of the
The statistical uncertainty of 2%, given above, refers
1000 to the standard deviation in the determination of α for
different velocities. If all data were purely shot noise
500
limited,wewouldexpectanuncertaintyassmallas0.1%.
The uncertainty in the measurement of the velocity
0
amounts to 1%, the voltages are known to 0.5%. The
0 2 4 6 8 10 12 14
lateral resolution of our interferometer is currently good
Voltage (kV)
to within ∼ 15nm. This amounts to less than two per-
FIG. 3: The Stark-deflection of the C70 fringes follows pre- centofthe maximumfringe shiftand itcorrespondsto a
ciselythequadraticvoltagedependence∆s∝U2/v2 ofEq.1: lateral force sensitivity of better than 10−14pN.
fullcircles: v¯=199m/s,σv =16%,opencirclesv¯=173m/s,
σv =13%, full squares: v¯= 109 m/s, σv =7%.
1600
where the velocity distribution function f (v) is ex-
v¯,σv
perimentally determined for several source settings, i.e. 1400
different mean velocities v¯. The fringe visibility function s
0
1
V(v) cannot be measured directly because of the finite er 1200
widthofthevelocitydistributionσ . Itisextractedfrom p
v s
the measured visibilities V(v¯,σv) using nt
u
o1000
C
V(v¯,σ )= dvf (v)V(v). (3)
v Z v¯,σv 800
The polarizability α is then obtained by fitting the
experimentally observedshift ∆s with the theoretical 0 5 10 15
exp
Voltage (kV)
shift ∆s (α).
th
We limit the maximum deflection voltage in the ex-
FIG. 4: Signal as a function of the deflection voltage for a
periment to 15kV since the fringes of slow (109m/s)
fixedposition of themask grating. Theopen circles are data
fullerenes are then already shifted by three entire pe-
points of C70 at a velocity of v¯ = 103.2m/s,σv = 7%. The
riods (Fig. 3) and dephasing effects become important. solidlineisthetheoreticalexpectationforα=108.5˚A3. The
Molecules of different velocities experience a different
dashed line is the envelope curve of the visibility function.
Starkdeflection,andthesummationoverthecorrespond- The dashed dotted lines illustrate upper and lower bounds
ingmoleculeinterferencepatternsinthedetectorwillre- resulting from small variations of the velocity distribution:
duce the totalfringe visibility forsufficiently largeveloc- ∆v¯=±1%,∆σv =±1% and typical uncertainties of the vis-
ityspreadsσ andwithincreasingvoltage(seeFig.4). In ibility: ∆V(v)=±3%.
v
our experiments σ (1/e2) ranges between 7% and 16%
v
formoleculesbetween100m/sand200m/s,respectively.
When all grating positions are fixed, the scanning de- The properties of the electric field are characterized
flection voltage can be used to sweep the molecular den- using a finite element code [24] which finds a value of
sity pattern across the third grating (Fig. 4). The data (E∇)E = (1.45 ± 0.08)· 1014V2/m3 for a voltage of
x
points are well described using the implicit functions of 10kV. The same code is used to determine the effective
the voltage for the expected visibility V¯ (α) and shift electrodelengthd =(4.73±0.1)cmwhichreplacesthe
th eff
∆s (α). It is interesting to note, that not only the shift real electrode length because of edge effects. The re-
th
but also the voltage dependent dephasing, which leads maining uncertaintiescomprise the accuracyinthe mea-
to a rather sharp decrease in the interference contrast, surement of the electrode’s contour and separation. All
may actually be used as tools for metrology. The fringe independent systematic uncertainties add up to a total
shifts are, however, less sensitive to additional random of 5.7%.
perturbations – such as vibrations, thermal radiation or In conclusion, we have demonstrated that a near-field
4
matter-wave interferometer is a promising tool for sen- possess a permanent electric dipole moment the system
sitively measuring the scalar polarizability. The three- behaves as an object with only an induced polarizability
grating design allows us to combine high spatial resolu- [28]. Therefore, dipole moments could already be mea-
tion and high molecular flux. sured in our current setup by only replacing the source.
As mentioned before, the velocity spread is a crit- Withrespecttofar-fieldinterferencemethodsthenear-
ical parameter, since it enters quadratically in Eq. 1. field scheme can be much extrapolated towards signifi-
However, in particular in pulsed molecular beam exper- cantly more complex particles. And even in its classical
iments, based on laser desorption and on pulsed photo- limit, as a multiplexing Stark-deflectometer it will still
ionizationthevelocityselectioncanreachvaluesassmall combine high resolutionwith high transmission. Finally,
as∆v/v ∼0.1%atacceptablesignalintensities[25],even it is interesting to note, that our apparatus may also
if the entire beam has still a finite temperature. This be regarded as a fast switch for molecular beams. The
way it is already possible to generate neutral beams of fine-structureimprintedonthe beam byits transmission
fullerenes, complex biomolecules [14, 26] up to polypep- throughthefirsttwogratingsallowsustolowerandraise
tides composed of 15 amino acids [25]. themoleculartransmissionwithkHz-frequencies. Inded-
The extension of our present setup to pulsed sources icated experiments it shouldbe possible to reacha mod-
and detectors is particularly interesting for instance for ulation amplitude of close to 100%.
investigatingthetemperaturedependenceofthemolecu-
lar polarizability. But also magnetic moments or triplet
lifetimes of fullerenes should thus be accessible.
Acknowledgments
Inbiomolecules,theconformationalvariationofpolar-
izabilities and electric dipole moments will be of inter-
est. If the molecules exhibit a finite electric or magnetic Our work is supported by the Austrian FWF through
dipolemoment,therandomorientationofcoldmolecules SFBF1505, STARTY177-2 and the European Commis-
will lead to a decrease of the visibility instead of a sim- sion under contract HPRN-CT-2002-00309. S. Deacha-
ple fringe shift [27]. As mentioned above, the amount of punya is supported through a Royal Thai Government
dephasing will then allow the extraction of a numerical scholarship. We thank H. Ulbricht for fruitful discus-
value for the moments. For hot molecular beams which sions.
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