A novel design for electric field deflectometry on extended molecular ...

IOP PUBLISHING

MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 19 (2008) 055801 (5pp)

doi:10.1088/0957-0233/19/5/055801

A novel design for electric field deflectometry on extended molecular beams André Stefanov1,3 , Martin Berninger2,4 and Markus Arndt2 1

Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria 2 Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria E-mail: [email protected]

Received 8 January 2008, in final form 27 February 2008 Published 2 April 2008 Online at stacks.iop.org/MST/19/055801 Abstract We discuss the optimal shape of a beam deflector with applications in electric susceptibility measurements on wide molecular beams. In contrast to the well-established ‘two-wire’ concept, which is optimized for beams with a small lateral extension, our design realizes a compact element that provides a high and homogeneous force field at moderate voltage for molecular beams with a large extension in the direction of deflection. Keywords: deflectometry, polarizability measurement

(Some figures in this article are in colour only in the electronic version)

Stark deflectometry experiments are based on the same idea, illustrated in figure 1(a): a beam of molecules with mass m and electrical polarizability α, or more generally with susceptibility χ interacting with a static electric field E experiences a lateral shift s due to a transverse force field F = χ(E∇)E: (E∇)E d d +L . (1) s(χ, v) = χ m vz2 2

1. Stark deflectometry versus Talbot–Lau and moiré deflectometry Since the beginning of molecular beam physics, deflection instruments have been of high relevance for determining the electric and magnetic properties of atoms and molecules. In recent years, an increasing number of publications have been devoted to the determination of electric susceptibilities. The knowledge about electromagnetic properties of clusters and molecules allows us to derive structural information or to compare competing computational models [1–8]. The internal electric properties have also been recognized as a valuable handle for efficient slowing, cooling and trapping of molecules in switched electric fields [9, 10] and in optical dipole force traps [11–16]. Finally, they also became important for the quantitative understanding of the molecule– grating interactions in various recent de Broglie interference experiments [17–20]. The electric susceptibility is most commonly measured by determining the deflection of a collimated molecular beam in an electrostatic field gradient [2, 21, 23–31]. All such

Here, vz is the longitudinal beam velocity, d is the length of the interaction region and L is the distance between the edge of the electrode and the detector. All precision deflection measurements aim at maximizing both the achievable signal and the sensitivity to tiny beam shifts. In order to further improve the position sensitivity, we have recently combined the Stark deflection concept with a micro-modulation of the molecular beam [22], as shown in figure 1(b). This microstructure of period g is imprinted through the action of two free-standing gratings, separated by the distance L. The first mask is designed to be an array of absorbers, such as a nanomechanical grating. The second grating may also be formed by a standing light wave [19]. The particle beam will then exhibit a density modulation of period g when observed at a distance L further downstream. Tiny shifts of the microstructure are well observable by

3 Present address: University of Bristol, Faculty of Physics, Bristol, BS8 1TL, UK. 4 Present address: University of Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria.

0957-0233/08/055801+05$30.00

1

© 2008 IOP Publishing Ltd

Printed in the UK

Meas. Sci. Technol. 19 (2008) 055801

A Stefanov et al

(a )

Figure 2. Simulation of the force field in a classical ‘two-wire’ configuration [36]. This design exhibits good vertical field homogeneity, but a linear force dependence in the horizontal direction. The white area shows the extension of our molecular beam.

(b)

deflection x and would vary by ±70% across the beam inside the TL-interferometer [36]. If we wanted to maintain the two-wire concept, a one percent force homogeneity could only be achieved if the electrode dimensions were increased by a factor of 100. The voltage would then have to be raised by a factor of 1000. For instance, for deflectometry experiments with thermal fullerene beams [22] this would amount to a voltage of more than 1 MV in order to achieve the required deflection force of 1014 V2 m−3 . Because of breakdowns, it would not be possible to experimentally reach such a high voltage. This strongly motivates a new compact electrode concept. In the following, we therefore describe a design which meets these stringent requirements. We then illustrate the field distribution using a finite element code and conclude by referring to a recent application of our electrode design to experiments with large molecules.

Figure 1. (a) Classical deflection experiment using a collimated beam and a two-wire field (e.g. [21]). (b) Deflectometry with a micro-modulated molecular beam, e.g. in a molecule interferometer [22].

recording the transmitted flux behind a third mask that is laterally shifted over the beam profile. Such a three-grating device can be operated in a quantum diffraction mode, as a Talbot–Lau interferometer (TLI) [18, 32], or in a classical moiré mode [33]. This concept permits us to work with a wide molecular beam and the fine overlaying fringe pattern allows us to reveal extremely small beam shifts, down to only a few nanometers. Any external force is therefore detectable with high sensitivity and the molecular susceptibility can be accurately determined [22, 34]. In order to achieve a precision of a few per cent in molecule metrology, the electric force field must be homogeneous to better than 1% over the typical beam extension. For our Talbot–Lau studies this amounts for instance to (x; y) ∼ (2000 µm ; 200 µm) [18]. A strong force in the x direction is required to achieve a sufficient beam deflection, whereas the force components in the other two directions should be negligible. We summarize this requirement as Fx /χ = (E∇)Ex → max, !

Fy /χ = (E∇)Ey = 0, !

Fz /χ = (E∇)Ez = 0.

2. Optimized deflector geometry Our new deflector is composed of two electrodes with an (x, z) mirror plane, as shown in figure 3. We first assume that the electrode shape is translation invariant along the molecular beam axis z, i.e. that Ez = 0 and ∂Ei /∂z = 0: ∂Ex ∂Ex ! + Ey = const, ∂x ∂y ∂Ey ∂Ey ! + Ey = 0, Fy /χ = Ex ∂x ∂y Fz /χ ≡ 0.

Fx /χ = Ex

(2) (3)

(5) (6) (7)

We further assume that the opposing surfaces are sufficiently flat and parallel that the electric field is dominantly oriented along the y direction. We therefore let Ex = 0, but we still keep all partial derivatives of Ex : using ∇ × E = 0 we find the differential equations for the vertical field component Ey : ∂Ey ∂Ex = Ey = const, (8) Fx /χ = Ey ∂y ∂x

(4)

A two-wire electrode configuration, as shown in figure 2, has commonly been used in many earlier experiments with narrow beams [21, 28, 35, 36]. It generates a high deflection force in a laterally tightly confined volume. Although the force is very uniform along y, it grows linearly in the direction of 2

Meas. Sci. Technol. 19 (2008) 055801

A Stefanov et al

Figure 3. The new deflector consists of two electrodes which are optimized for maximum force uniformity over a horizontally extended molecular beam. The outline of the electrode is described by s(x) = (ax + b)−1/2 (equation (12)) between the points A and B.

Fy /χ = Ey

∂Ey = 0. ∂y

Figure 4. Finished deflector made of stainless steel.

(9)

From equation (9) we see that Ey is independent of the vertical position y, i.e. Ey (x, y) = Ey (x). This is later verified in the numerical simulations as well. Since we are searching for the equipotential surface s(x) that generates such a field we locally approximate the electrodes by successive plane–parallel capacitors separated by s(x). This is again well justified by the numerical results below: Ey (x) V /s(x),

(10)

2

Fx V ∂ V V ∂s(x) =− 3 = const. χ s(x) ∂x s(x) s (x) ∂x

(11) Figure 5. Zoom into the right half of the deflector. The molecular beam is centered on the deflector coordinates (7; 0) mm. The thick (red) line encircles the region over which the deflection force amounts to F /χ = 1.53 × 1014 V2 m−3 with a maximum variation of ±0.5%.

The outline of the electrode is then given by the following solution: −1/2 2Fx −2 s(x) = · x + s (0) . (12) χV 2

a typical three-grating deflection experiment, (x; y) = (2; 0.2) mm. The force field in the y direction Fy /χ is proportional to (E∇)Ey = 4 × 1012 V2 m−3 . This is about 30 times smaller than the force in the x direction and the condition of equation (3) is rather well fulfilled. We note that in the region of interest |Ex ∂Ex /∂x| < 2 × 1010 V2 m−3 . The small magnitude of this number shows that we can safely neglect its contribution to the total force in the x direction. Instead, Fx is dominated by the term Ey · ∂Ex /∂y. It is constant although both factors vary in a correlated way by about 10% individually.

Fx changes sign at the transition from the left to the right half of the electrode. The force therefore always points toward the field maximum, i.e. to the symmetry center.

3. Force field simulations In order to get a quantitative prediction for the example of a specific deflection experiment inside a Talbot–Lau interferometer [22, 34], we compute the force distribution using the finite element code FEMLAB. For that purpose s(x) is approximated by a set of Bezier curves. We choose a minimum separation of 4.2 mm between the grounded top electrode and the bottom electrode at +10 kV. The width of the electrode is chosen to be 4 cm, while its length is set to 5 cm (see figure 4). The numerical analysis in figure 5 shows that we achieve a force of Fx /χ = (E∇)Ex = 1.53×1014 V2 m−3 which is most homogeneous at the coordinates (x; y) = (7; 0) mm. The field uniformity is actually better than 1% over a rectangular region larger than (x; y) = (3; 2) mm. The full one-percent force deviation region is even significantly more extended, as indicated by the thick (red) line. Both areas well exceed the above-mentioned dimensions of the molecular beam in

4. Experimental implementation We have implemented this design in a recent matter wave experiment [22]. As shown in figure 4, we have additionally rounded the edges at the entrance and the exit face with a 2.5 mm radius of curvature, in order to minimize the probability of spark discharges. To stay within the simple formalism derived further above, we may compute the line integral of the true force field along the beam trajectory and then replace the rounded real electrode numerically by an electrode pair of effective 3

Meas. Sci. Technol. 19 (2008) 055801

A Stefanov et al

length deff with straight edges that creates the same field [(E∇)Ex ]max = 1.53 × 1014 V2 m−3 , the same momentum transfer and the same beam deflection: ∞

−∞

[(E∇)Ex ](z) dz = deff [(E∇)Ex ]max .

References [1] Baker J, Fowler P W, Lazzeretti P, Malagoli M and Zanasi R 1991 Structure and properties of C70 Chem. Phys. Lett. 184 182–6 [2] Compagnon I, Antoine R, Broyer M, Dugourd Ph, Lerme J and Rayane D 2001 Electric polarizability of isolated C70 molecules Phys. Rev. A 64 025201–4 [3] Guha S, Menéndez J, Page J B and Adams G B 1995 Empirical bond polarizability model for fullerenes Phys. Rev. B 53 13106–14 [4] Jonsson D, Norman P, Ruud K, Agren H and Helgaker T 1998 Electric and magnetic properties of fullerenes J. Chem. Phys. 109 572–7 [5] Matsuzawa N and Dixon D A 1992 Semiempirical calculations of the polarizability and second-order hyperpolarizability of C60 , C70 , and model aromatic compounds J. Phys. Chem. 96 6241–7 [6] Shanker B and Applequist J 1994 Polarizabilities of fullerenes C20 through C240 from atom monopole–dipole interaction theory J. Phys. Chem. 98 6486–9 [7] Shuai Z and Brédas J L 1992 Electronic structure and nonlinear optical properties of the fullerenes C60 and C70 : a valence-effective-Hamiltonian study Phys. Rev. B 46 16135–41 [8] Willaime F and Falicov L M 1993 Spin-density wave, charge density wave, and polarizabilities of C60 in the Pariser–Parr–Pople model J. Chem. Phys. 98 6369–76 [9] Bethlem H L, Berden G and Meijer G 1999 Decelerating neutral dipolar molecules Phys. Rev. Lett. 83 1558 [10] Bethlem H L, Berden G, Van Roij A J A and Meijer G 2000 Trapping neutral molecules in a traveling potential well Phys. Rev. Lett. 84 5744–7 [11] Barker P F and Shneider M N 2001 Optical microlinear accelerator for molecules and atoms Phys. Rev. A 64 033408–9 [12] Fulton R, Bishop A I and Barker P F 2004 Optical Stark decelerator for molecules Phys. Rev. Lett. 93 243004–4 [13] Grimm R, Weidemüller M and Ovchinnikov Y B 2000 Optical dipole traps for neutral atoms Adv. At. Mol. Opt. Phys. 42 95–170 [14] Takekoshi T, Patterson B M and Kinze R J 1998 Observation of optically trapped cold cesium molecules Phys. Rev. Lett. 81 5105–8 [15] Takekoshi T, Yeh J R and Knize R J 1995 Quasi-electrostatic trap for neutral atoms Opt. Commun. 114 421 [16] Weidemüller M and Zimmermann C (ed) 2003 Interactions in Ultracold Gases (New York: Wiley) [17] Arndt M, Nairz O, Voss-Andreae J, Keller C, Van der Zouw G and Zeilinger A 1999 Wave-particle duality of C60 molecules Nature 401 680–2 [18] Brezger B, Hackermüller L, Uttenthaler S, Petschinka J, Arndt M and Zeilinger A 2002 Matter-wave interferometer for large molecules Phys. Rev. Lett. 88 100404 [19] Gerlich S et al 2007 A Kapitza–Dirac–Talbot–Lau interferometer for highly polarizable molecules Nature Phys. 3 711–5 [20] Hackermüller L, Uttenthaler S, Hornberger K, Reiger E, Brezger B, Zeilinger A and Arndt M 2003 Wave nature of biomolecules and fluorofullerenes Phys. Rev. Lett. 91 90408 [21] Antoine R, Dugourd Ph, Rayane D, Benichou E, Broyer M, Chandezon F and Guet C 1999 Direct measurement of the electric polarizability of isolated C60 molecules J. Chem. Phys. 110 9771–2 [22] Berninger M, André S, Deachapunya S and Arndt M 2007 Polarizability measurements in a molecule near-field interferometer Phys. Rev. A 76 013607

(13)

We find an effective electrode length of deff = 4.73 cm. Slight discrepancies in the deflection between the real and the effective electrode were estimated to be of the order of only 0.6%. Edge effects in the y directions are negligibly small. The numerically optimized shape of the electrodes was exported to a CAD program and realized as a stainless steel prototype on a CNC milling machine. The supporting frame of the deflector was made of machinable ceramic. The optimal beam path through the electrode system was defined to better than 100 µm by precisely positioned holes in the ceramic walls. The field value is then defined by our knowledge of the geometry and the applied voltage. The electrodes in the experiment had a gap distance of 4.3 ± 0.1 mm and generated a field of (E∇)Ex = (1.45 ± 0.08) × 1014 V2 m−3 at a potential of 10 kV. Taking into account edge effects, the total systematic uncertainty of the force field across the molecular beam diameter can be estimated to be better than 4%. Inserting this electrode into our Talbot–Lau matter wave interferometer, we were able to derive the static polarizability of the fullerenes C60 and C70 [36] as well as tetraphenylporphyrin and a derivative thereof [34], with beams up to 2 mm wide. The micro-modulation had a period of 990 nm and allowed detection of beam shifts down to 10 nm. The total error of the present measurements is between 5% and 10%, depending on the details of the experiment.

5. Conclusion The deflector itself is very adaptable for a wide range of beam deflection studies, with the clear potential to increase the experimental accuracy to below 1% for beams with improved velocity selection, position control and also using atoms with known polarizabilities for an improved field calibration. Such an accuracy is expected to be a natural best possible value in practice for molecules with a rapidly changing conformation and varying scalar polarizability on this level. Future experiments will aim at determining for instance the temperature dependence of the molecular electric susceptibility. A recent feasibility study [37] has also shown the possibility to physically separate sequence isomers of polypeptides and possibly even metallic from semiconducting carbon nanotubes inside a three-grating-based deflectometer using our new electrode design. Generally, such electrodes can be used in all applications where a homogeneous (E∇)E force field is needed.

Acknowledgments We acknowledge the support by the Austrian FWF through SFB F1505, and STARTY177-2, as well as by the European Commission through Integrated Project Qubit Applications (QAP). 4

Meas. Sci. Technol. 19 (2008) 055801

A Stefanov et al

[23] Antoine R, Rayane D, Allouche A R, Aubert-Frecon M, Benichou E, Dalby F W, Dugourd Ph and Broyer M 1999 Static dipole polarizability of small mixed sodium–lithium clusters J. Chem. Phys. 110 5568–77 [24] Antoine R, Compagnon I, Rayane D, Broyer M, Dugourd Ph, Breaux G, Hagemeister F C, Pippen D, Hudgins R R and Jarrold M F 2002 Electric susceptibility of unsolvated glycine-based peptides J. Am. Chem. Soc. 124 6737–41 [25] Bonin K and Kresin V 1997 Electric-Dipole Polarizabilities of Atoms, Molecules and Clusters (Singapore: World Scientific) [26] Chamberlain G and Zorn J 1963 Alkali polarizabilities by the atomic beam electrostatic deflection method Phys. Rev. 129 677–80 [27] Compagnon I, Hagemeister F C, Antoine R, Rayane D, Broyer M, Dugourd Ph, Hudgins R R and Jarrold M F 2001 Permanent electric dipole and conformation of unsolvated tryptophan J. Am. Chem. Soc. 123 8440–1 [28] Hall W and Zorn J 1974 Measurement of alkali-metal polarizabilities by deflection of a velocity-selected atomic beam Phys. Rev. 10 1141–4 [29] Moro R, Rabinovitch R, Xia C and Kresin V V 2006 Electric dipole moments of water clusters from a beam deflection measurement Phys. Rev. Lett. 97 123401

[30] Rayane D et al 1999 Static electric dipole polarizabilities of alkali clusters Eur. Phys. J. D 9 243–8 [31] Rayane D, Antoine R, Dugourd Ph, Benichou E, Allouche A R, Aubert-Frécon M and Boyer M 2000 Polarizability of KC60 : evidence for potassium skating on the C60 surface Phys. Rev. Lett. 84 1962–5 [32] Clauser J and Li S 1997 Generalized Talbot–Lau atom interferometry Atom Interferometry ed P R Berman vol 11 (New York: Academic) pp 121–51 [33] Oberthaler M K, Bernet St, Rasel E M, Schmiedmayer J and Zeilinger A 1996 Inertial sensing with classical atomic beams Phys. Rev. A 54 3165–76 [34] Deachapunya S, André S, Berninger M, Ulbricht H, Reiger E, Doltsinis N L and Arndt M 2007 Thermal and electrical properties of porphyrin derivatives and their relevance for molecule interferometry J. Chem. Phys. 126 164304 [35] Ramsey N F 1956 Molecular Beams (Oxford: Oxford University Press) [36] Tikhonov G, Wong K, Kasperovich V and Kresin V V 2002 Velocity distribution measurement and two-wire field effects for electric deflection of a neutral supersonic cluster beam Rev. Sci. Instrum. 73 1204–11 [37] Ulbricht H, Berninger M, Deachapunya S, Stefanov A and Arndt M 2008 Gas phase sorting of fullerenes, polypeptides and carbon nanotubes Nanotechnology 19 045502

5