Acoustic funnel and buncher for nanoparticle injection
Zheng Li, Liangliang Shi, Lushuai Cao, Zhengyou Liu, Jochen Küpper
aa r X i v : . [ phy s i c s . a pp - ph ] M a y Acoustic funnel and buncher for nanoparticle injection
Zheng Li,
1, 2, ∗ Liangliang Shi, † Lushuai Cao, ‡ Zhengyou Liu, and Jochen Küpper § Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany The Hamburg Center for Ultrafast Imaging, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Department of Physics, Wuhan University, 430072 Wuhan, China Department of Physics, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
Acoustics-based techniques are investigated to focus and bunch nanoparticle beams. This allows to overcomethe prominent problem of the longitudinal and transverse mismatch of particle-stream and x-ray-beam size insingle-particle/single-molecule imaging at x-ray free-electron lasers (XFEL). It will also enable synchronizedinjection of particle streams at kHz repetition rates. Transverse focusing concentrates the particle flux to thesize of the (sub)micrometer x-ray focus. In the longitudinal direction, focused acoustic waves can be usedto bunch the particle to the same repetition rate as the x-ray pulses. The acoustic manipulation is based onsimple mechanical recoil effects and could be advantageous over light-pressure-based methods, which rely onabsorption. The acoustic equipment is easy to implement and can be conveniently inserted into current XFELendstations. With the proposed method, data collection times could be reduced by a factor of . This workdoes not just provide an efficient method for acoustic manipulation of streams of arbitrary gas-phase particles,but also opens up wide avenues for acoustics-based particle optics. X-ray free-electron lasers enable single-particle and single-molecule imaging by x-ray diffraction [1], due to the unprece-dented brightness and femtosecond pulse duration. As the par-ticle stream enters the vacuum chamber, transverse expansionis inevitable for freely moving particles due to the pressuredifference. At present, one of the key bottlenecks in single-particle imaging at XFELs is the large size of aerodynami-cally focused particle streams, often of a few tens of microme-ters [2, 3] compared to the small size of the 100 nm-diameterx-ray beam. Furthermore, in the longitudinal direction the par-ticles passing between the pulses are also not intercepted. Thismismatch results in low sample delivery efficiency, only aboutone in particles are intercepted in the case of a 100 µmparticle beam moving at 100 m/s across a 100 nm x-ray beamat a 1 kHz repetition rate. As a result, many samples, whichare often precious, are wasted, and days of data collectionare often required in order to obtain only a few hundred orperhaps thousand high-quality diffraction patterns at an x-raypulse repetition rate of some kHz, whereas > patternsare required for atomic-resolution imaging [4].Different means to enhance the interception rate of parti-cles by the x-ray pulses through transverse focusing are con-sidered, such as improved aerodynamic collimation [5–7] orthe focusing with laser traps [8]. Furthermore, bunching [9],i. e., longitudinal focusing, with spatial periods that match therepetition rate of x-ray pulses could be utilized to further im-prove sample use. Suppose the particles stream was trans-versely compressed to 1 µm and bunched to millimeter sizewith the same frequency as the repetition rate of x-ray pulsesin the longitudinal direction: compared to the typical param- ∗ [email protected]; Present Address: School of Physics, Peking University,Beijing, China. † Present address: Paul Scherrer Institut PSI, CH-5232 Villigen, Switzerland ‡ Z. Li, L. Shi and L. Cao contributed equally to this work. § eters given above, data collection time and sample use wouldbe reduced by a factor of .Here, we propose that the longitudinal and transverse ma-nipulation of the particle stream can be realized by an acous-tic funnel and a buncher as sketched in Fig. 1. In gas flows,the deviation from continuum behavior is quantified by theKnudsen number, Kn = Λ /H , where Λ is the mean free pathand H is a characteristic length scale, which can be taken asthe width between transducer and reflector. Kn > corre-sponds to ballistic molecular behavior of free molecular flow, . ≤ Kn ≤ is known as the transition regime, and for Kn . . a continuum hydrodynamic description is possi-ble. We focus on the case of Kn < . , for which the con-ventional picture of acoustic waves in continuum media isvalid [10]. For helium gas at T = 5 K the mean free pathis
Λ = k B T / √ πσ p = 2 mm, with the size of the heliumatom σ = 280 pm and the pressure p = 10 − mbar, or simi-larly for p = 5 × − mbar at room temperature. The widthof the standing wave resonator is H = ( n + 1 / λ = 2 . cmwith an acoustic wave of wavelength λ = 5 mm and frequency ν = 26 kHz. In the following, we present the theory for thetransverse and longitudinal manipulation with standing andtraveling acoustic waves, respectively.As illustrated in Fig. 1, the acoustic funnel is made of twoorthogonal half-wavelength cavities formed by transducersand specular reflectors in the transverse direction. The two1D cavities are set up to overlap in the center of the particlebeam. Since the focusing in the transverse x, y directions issimilar, we firstly consider the Gor’kov potential U ( x, y ; t ) forfocusing in the x, y direction [12–14] U = 16 πR Ic (cid:20) f (cos kx + cos ky + 2 cos kx cos ky ) × sin ωt − f (sin kx + sin ky ) cos ωt (cid:21) (1)with f = 1 − ( ρc ) / ( ρ c ) and f = 2( ρ − ρ ) / (2 ρ + ρ ) . I and k are the intensity and wave number of the acoustic field, FIG. 1. Front view (a), with three-dimensional perspective, of asketch of the setup, including the configuration of the acoustic fun-nel and buncher as well as the detection window (DW), which isequipped with white light (WL) illumination and camera. The acous-tic funnel is formed by two orthogonal standing wave resonators inthe x, y directions, each consisting of a transducer (T) and a reflector(R), and the particle flow is along z direction. A top view of the cre-ated acoustic potential is shown in (b). The coordinates refer to thecenter of the mechanical setup of transducers and reflectors. Poten-tial minima are characterized by an oval shape and the center of oneis marked by a red dot at − ( λ/ , λ/ , corresponding to the posi-tion where the focusing experiment is performed. Below the funnel,the acoustic buncher is formed by two tilted transducers that emitsynchronized acoustic waves. The acoustic wave is transversely fo-cused by a conical cavity with a pinhole [11]. The upper chamberis filled with helium gas, at a pressure of − mbar, as the acousticcoupling medium. The particle stream enters from the top and movesdownward. respectively, R is the radius of the particle, c and ρ are thespeed of sound in and the density of the coupling medium,and c and ρ are the speed of sound in and the density of theparticle. Due to the fast velocity of the nanoparticles, we keepthe form of Gor’kov force with temporal modulation [12]. Aswill be shown below, the exact form of static Gor’kov forcerelies on the condition that the characteristic frequency of par-ticle motion has to be much lower than that of the acousticwave, such that the particles have stable trajectories, and thiscondition can be well fulfilled in our scheme. We assume mass and radius of the particle as m = 3 × − kg and R = 100 nm, which resembles typical biologi-cal sample particles, such as virus particle. (1) corresponds tothe force from the potential of an eigen mode that has a mini-mum at the center of the cavity r = 0 [15–17]. Assuming theparticle has, at least, one symmetry axis and the longitudinalmotion is parallel to that axis, there is no deflecting force inthe transverse direction [18]. Thus the Brownian motion is thedominant mechanism of transverse dispersion of the particlebeam. Denoting the transverse velocity as v y = ˙ y , the equa-tion of motion for the Brownian motion in Gor’kov potentialis m ˙ v y + βv y = F B ( t ) + F G ( y, t ) v y (0) = 0 , y (0) = 0 , (2)where F B ( t ) is the force of Brownian collision and F G ( y, t ) isthe Gor’kov force. For low pressure, p . − mbar, heliumas the coupling medium, and a Knudsen number close to thetransition regime, the friction coefficient β can be expressedas β = 4 πR ρ r k B Tm a . (3)We can linearize the Gor’kov force around − ( λ/ , λ/ as F G ( y, t ) = − πIR ( kR ) c "(cid:18) f + 12 f (cid:19) − cos 2 ωt × (cid:18) f − f (cid:19) y = − Gy (cid:18) HG cos 2 ωt (cid:19) . (4)The motion in the x -direction is the same, since the linearizedGor’kov force F G ( x, t ) can be obtained by replacing y with x . The oscillating term in the Gor’kov force that is propor-tional to cos 2 ωt can possibly induce parametric resonancesand drive particles away from the equilibrium position of thepotential. However, it can be shown that the parametric reso-nance can be safely avoided in our case, due to a large differ-ence between the frequencies of particle oscillation and acous-tic wave: Rewriting (2) approximately in the form of a Math-ieu equation ¨ y + βm ˙ y + Gm y (1 + HG cos 2 ωt ) = 0 . (5)and denoting Ω = p G/m as the characteristic frequency ofparticle oscillation, the particle trajectory is found as y ( t ) = e − βt m C C "(cid:18) Ω ω (cid:19) − (cid:18) β mω (cid:19) , − H G (cid:18) Ω ω (cid:19) , ωt + C S "(cid:18) Ω ω (cid:19) − (cid:18) β mω (cid:19) , − H G (cid:18) Ω ω (cid:19) , ωt , (6) ■(cid:0)✁ m ✁✭✂✄☎☎s✆✭✝✞✟✲✠✡☛ ✲☛✡☞ ☛✡☛ ☛✡☞ ✠✡☛❛☛✡☛☛✡☞✠✡☛q ☛✡☛☛✡☞✠✡☛✠✡☞ FIG. 2. Stability diagram according to (5) with β = 0 . The regionwith a purely imaginary characteristic exponent of Im[ µ ( a, q )] > permits stable trajectories according to the Mathieu equation, seecolor map, and the unstable region is left white. The parameters forour case correspond to the solid lines a = (Ω /ω ) = (2 G/H ) q . where C ( a, q, ν ) and S ( a, q, ν ) are even and odd Mathieu func-tions. Rigorous theory of Mathieu equations gives the stableregime of the particle trajectory with β = 0 [19, 20], seeFig. 2. In this parameter space the particles oscillate trans-versely with limited amplitudes that do not grow exponen-tially. A wide range of ratios between particle-oscillation andacoustic-wave frequencies provide stable trajectories. Therequired condition can be conveniently fulfilled even with-out friction, e. g., in our case ( a, q ) ≃ (0 . , . × − ) .Variational analysis demonstrated that the friction can furtherwiden the permitted stable regime according to the Mathieuequation [21, 22], since it physically suppresses the oscilla-tion amplitude of particle trajectory.Computations following (5) show converging trajectories tothe center of the harmonic potential. In the absence of para-metric resonances, the particle trajectories must converge tothe focused area. Similar to the case of a pure harmonic poten-tial the temporal factor in the Gor’kov force could be approxi-mately integrated out [12]. Based on the stability analysis, theparticle’s velocity is v y ( t ) = − βm y + 1 m ˆ t F B ( ζ ) dζ + 1 m F G ( y ) t . (7)The corresponding Fokker-Planck equation [23–25] for thetransverse distribution of the particles f ( y, y , t ) can thus beobtained, for the linearized Gor’kov force, as ∂∂t f = Gβ ∂∂y ( yf ) + D ∂ ∂y f . (8)From the Fokker-Planck equation, the temporal evolution of the transverse particle positions are obtained as f ( y, y , t ) = " G πβD (1 − e − Gβ t ) / × exp − G βD (cid:16) y − y e − Gβ t (cid:17) − e − Gβ t . (9)This yields the minimal width of the particle stream as w min = r k B TG . (10)Given an initial width w , the transverse distribution function f ( y, y , t ) in (9) can be convoluted as f ( y, t ) = ˆ ∞−∞ f ( y, y , t ) w ( y ) dy w ( y ) = s ln 2 πw e − ln 2 y /w , (11)which gives the temporal evolution of particle stream f ( y, t ) = P s π Q e − Gβ t + p ln 2 /w (12) × exp − Q − Q e − Gβ t Q e − Gβ t + p ln 2 /w ! y , where P ( t ) = p G ln 2 / (2 πw βD (1 − e − Gt/β )) , and Q ( t ) = G/ (2 βD ) · / (1 − e − Gt/β ) . The temporal evolution obtained for w = 100 µm is pre-sented in Fig. 3. The particle beam is transversely compressedto a width of µm, approaching the size of the XFEL beam.We show the temporal evolution of particle number densitydistribution determined from (9) in Fig. 3(a), and from numer-ical simulations in Fig. 3(b) and (c).The acoustic buncher relies on the period force imposed bythe traveling wave resulting from tilted transducers, see Fig. 1.Suppose the two transducers radiate synchronously with thesame phase, then the transverse force is zero and only a forcein the longitudinal direction remains. In our case, the parti-cles move with a longitudinal velocity of v z ∼ m/s, andthe buncher imposes a force field that has sufficiently shortlongitudinal interaction length, i. e., the particle transit time ∆ t = l/v z is much shorter than the period of the acousticwave. Since the acoustic pressure variation does not affectthe particle for a full cycle, the particle only experience atransient force. This leads to an acoustic force that is pro-portional to the first order of the sinusoidal modulation of theplane acoustic wave. Assuming the acoustic pressure to be p = p sin( ~k · ~r − ωt + φ ) , it takes a form p = p sin (cid:20) ~k · (cid:16) ~r ( t ) + ~R (cid:17) − ωt + φ (cid:21) , (13) FIG. 3. (a) Temporal evolution of a particle distribution with an ini-tial width (waist) w = 100 µm in an acoustic wave with frequency ν = 26 kHz and an intensity of I = 1 W/cm . The final width isconsistent with the minimal width w min = 7 . µm, determined from(10). Particle number density distributions are plotted at (b) 0 µs and(c) 70 µs from numerically simulated dynamics using the 2D poten-tial with explicit time dependence in (1). The root mean squareddeviation from the center ( − λ/ , − λ/ is 100 µm and 6 µm at 0 µsand 70 µs, respectively. for ~r = ~r ( t ) + ~R on the surface of a particle at position ~r ( t ) with radius R, where ~k is the wave vector and φ is an arbitraryphase. Under this assumption, the acoustic force exerted onthe particle is f z = ‹ pdS = ˆ π dφ ˆ π dθ sin θR p × sin h kR cos θ − ωt + ~k · ~r ( t ) + φ i = 4 πR sin kRk sin (cid:16) ωt − ~k · ~r ( t ) − φ (cid:17) . (14)Because of the narrow width of the interaction area of thebuncher, and the . nm size of the particles, i. e., kR ≪ ,the particle scattering effect is suppressed, and the force canbe further approximated as f z ≃ p S sin( ωt − θ ) , where θ is a constant phase factor, left to be chosen, and S is the sur-face area of the particle. In general, we write the effectivelongitudinal force as f z = F sin ωt .Considering an acoustic wave with ν = 1 kHz and an in-duced relative pressure variation of ∆ p = 2 × − mbar,well below the pressure in the chamber, we obtain a force F = 10 − pN. The particles experience a periodic velocity modulation with respect to their entrance time into the inter-action region of length d , yielding mv − E ≈ F d sin ωt with the particle velocity v and kinetic energy E = mv at the entrance of the interaction region. The length d ofthe particle-wave-interaction region is chosen such that theparticle experiences the force over only / of the acous-tic wave period. A conical cavity with a pin-hole can focusthe acoustic wave to a length on the order of λ/ in the nearfield [11]. Considering the wavelength of the 1 kHz acousticwave, λ = 11 cm, we choose the length of interaction regionto be d = 1 cm, which is experimentally feasible. Thus wehave approximately v = v (cid:2) F d/ E ) sin ωt (cid:3) . Assuming particles drift for a distance l after leaving theinteraction region and arrive at the end of the buncher at time t , we have t = t + lv ≃ t + lv (cid:18) − F d E sin ωt (cid:19) . (15)With an initial number density n and the continuity condition n dt = n dt , the modulated number density n at t can beexpressed as n = n + ∞ X k =1 a k cos (cid:2) k ( ωt − Θ) (cid:3) + b k sin (cid:2) k ( ωt − Θ) (cid:3) with (16) a k = n π ˆ Θ+ π Θ − π cos (cid:2) k ( ωt − X sin ωt ) (cid:3) d ( ωt )= 2 n J k ( kX ) b k = 0 for k = 1 , , . . . , where Θ = lω/v , X = F dlω/ (2 E v ) , and J k ( x ) is theBessel function of k -th order. We consider the fundamentalharmonic n = n + 2 n J ( X ) sin( ωt − Θ) . (17)The degree of bunching is determined by the bunching param-eter X . The frequency of the traveling wave can be conve-niently set as the repetition rate of x-ray pulses.After the particle stream passes the interaction region oflength d it can continue into the next chamber, see Fig. 1,and drifts for a distance l to the interaction point. Assum-ing I = 1 W/cm , ν = 1 kHz, d = 1 cm, v = 100 m/s,and that the cavities are tilted by Ψ = π/ , the degree ofbunching is maximized as the Bessel function J ( X ) reachesits maximum at X ≃ . , which corresponds to a drift length l = 87 cm.We numerically simulate the bunching process using par-ticle tracing methods [26]. In the simulation, the buncher isoperated such that a 6 cm long packet of molecules with a lon-gitudinal velocity of 100 m/s and a velocity spread of 1 m/s en-ters the acoustic buncher. The impulse by a force of − pNacting on particle of × − kg for ∆ t = d/v ≃ . mscan modulate the velocity by ∆ v ≃ . m/s. This can beused as the criterion to choose the acoustic pressure, since themodulation must be similar to that of the velocity spread of ✲(cid:0)✲✁✲✂✲✄✥✄✂✁(cid:0)✲✥☎✥✁ ✲✥☎✥✂ ✲✥☎✥✄ ✥ ✥☎✥✄ ✥☎✥✂ ✥☎✥✁ D ✈ ③✆✝✞✟✠ D ✡ ☛☞✌ t✍✥☎✥ ☞✎t✍✂☎✏ ☞✎t✍✏☎✥ ☞✎ FIG. 4. The calculated longitudinal phase-space distribution of theparticles is given at the entrance of the buncher, t = 0 . ms, and thatin the detection region at t = 5 . ms as well as an intermediate time t = 2 . ms demonstrating the phase-space rotation; all distributionsrelative to the phase-space position of the synchronous particle. the particle beam. The calculated distribution at t = 5 ms,the time at which the longitudinal spatial focus is obtaineddownstream of the buncher, is shown in Fig. 4. The longitudi-nal phase space distribution is relative to the position in phasespace of the “synchronous particle” [9]. In the particular situa-tion depicted in Fig. 4, the molecular packet has a longitudinalfocus with a length of about 3 mm some 53 cm after the end ofthe buncher. The longitudinal focal length is consistent withour simplified model with a single velocity and infinitely short interaction region.We have proposed an acoustic method to manipulate andcompress particle streams by transverse and longitudinal fo-cusing, which enables high-efficiency particle delivery, forinstance, for single-particle diffractive imaging experimentswith sub-µm-focus x-ray beams. This can substantially reducethe data collection time in such XFEL based imaging experi-ments. The effective manipulation of particle streams basedon acoustic waves could be applied to wider scope of molecu-lar beam experiments, such as matter-wave-interference withlarge molecules [27] as well as applications to fast highly colli-mated beams [5]. Furthermore, this work does not just providean efficient method for acoustic manipulation of gas-phase-particle streams, but also sheds light on the application of thevast particle-optics techniques from accelerator physics to thefield of acoustics, e. g., such as particle bunching by the trav-eling wave from analogues to iris-loaded waveguides.The authors gratefuly acknowledge stimulating discussionswith R. J. Dwayne Miller, Oriol Vendrell, Nikita Medvedev,Sheng Xu, Ludger Inhester, and Henry N. Chapman.This work has been supported by a Peter Paul EwaldFellowship of the Volkswagen Foundation, by the Euro-pean Research Council under the European Union’s Sev-enth Framework Programme (FP7/2007-2013) through theConsolidator Grant COMOTION (ERC-614507-Küpper), bythe Clusters of Excellence “Center for Ultrafast Imaging”(CUI, EXC 1074, ID 194651731) and “Advanced Imaging ofMatter” (AIM, EXC 2056, ID 390715994) of the DeutscheForschungsgemeinschaf, and by the Helmholtz Gemeinschaftthrough the “Impuls- und Vernetzungsfond”. [1] John C H Spence and Henry N Chapman, “The birth of a newfield,” Phil. Trans. R. Soc. B , 20130309–20130309 (2014).[2] Max F Hantke, Dirk Hasse, Filipe R N C Maia, Tomas Ekeberg,Katja John, Martin Svenda, N Duane Loh, Andrew V Martin,Nicusor Timneanu, Daniel S D Larsson, Gijs van der Schot,Gunilla H Carlsson, Margareta Ingelman, Jakob Andreas-son, Daniel Westphal, Mengning Liang, Francesco Stellato,Daniel P Deponte, Robert Hartmann, Nils Kimmel, Richard AKirian, M Marvin Seibert, Kerstin Mühlig, Sebastian Schorb,Ken Ferguson, Christoph Bostedt, Sebastian Carron, John DBozek, Daniel Rolles, Artem Rudenko, Sascha Epp, Henry NChapman, Anton Barty, Janos Hajdu, and Inger Andersson,“High-throughput imaging of heterogeneous cell organelleswith an x-ray laser,” Nature Photon. , 943–949 (2014).[3] S Awel, R A Kirian, M O Wiedorn, K R Beyerlein, N Roth,D A Horke, D Oberthür, J Knoska, V Mariani, A Mor-gan, L Adriano, A Tolstikova, P L Xavier, O Yefanov, An-drew Aquila, Anton Barty, S Roy-Chowdhury, M S Hunter,D James, J S Robinson, U Weierstall, A V Rode, S Bajt,Jochen Küpper, and Henry N Chapman, “Femtosecond x-raydiffraction from an aerosolized beam of protein nanocrystals,”J. Appl. Crystallogr. , 133–139 (2018).[4] Anton Barty, Jochen Küpper, and Henry N. Chap-man, “Molecular imaging using x-ray free-electron lasers,”Ann. Rev. Phys. Chem. , 415–435 (2013).[5] R. A. Kirian, S. Awel, N. Eckerskorn, H. Fleckenstein,M. Wiedorn, L. Adriano, S. Bajt, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, M. Domaracky, M. Heymann,D. A. Horke, J. Knoska, M Metz, A Morgan, D Oberthuer,N Roth, T Sato, P L Xavier, O. Yefanov, A. V. Rode, JochenKüpper, and Henry N. Chapman, “Simple convergent-nozzleaerosol injector for single-particle diffractive imaging with x-ray free-electron lasers,” Struct. Dyn. , 041717 (2015).[6] Nils Roth, Salah Awel, Daniel Horke, and Jochen Küpper,“Optimizing aerodynamic lenses for single-particle imaging,”J. Aerosol Sci. , 17–29 (2018).[7] Daniel A Horke, Nils Roth, Lena Worbs, and Jochen Küp-per, “Characterizing gas flow from aerosol particle injectors,”J. Appl. Phys. , 123106 (2017).[8] Niko Eckerskorn, Richard Bowman, Richard A. Kirian, SalahAwel, Max Wiedorn, Jochen Küpper, Miles J. Padgett, Henry N.Chapman, and Andrei V. Rode, “Optically induced forces im-posed in an optical funnel on a stream of particles in air or vac-uum,” Phys. Rev. Applied , 064001 (2015).[9] Sebastiaan Y T van de Meerakker, Hendrick L Bethlem, Nico-las Vanhaecke, and Gerard Meijer, “Manipulation and controlof molecular beams,” Chem. Rev. , 4828–4878 (2012).[10] N. G. Hadjiconstantinou, “Sound wave propaga-tion in transition-regime micro- and nanochannels,”Phys. Fluids , 802 (2002).[11] Katsuhiro Sasaki, Morimasa Nishihira, and Kazuhiko Imano,“Low-frequency air-coupled ultrasonic system beyond diffrac-tion limit using pinhole,” Jap. J. Appl. Phys. , 4560 (2006).[12] L. P. Gor’kov, “On the forces acting on a small particle in an acoustical field in an ideal fluid,”Doklady Akademii Nauk SSSR , 88 (1961).[13] Stefano Oberti, Adrian Neild, and Jürg Dual, “Manipulationof micrometer sized particles within a micromachined fluidicdevice to form two-dimensional patterns using ultrasound,” J.Acoust. Soc. Am. , 778–785 (2007).[14] N. Li, A. Kale, and A. C. Stevenson, “Axial acous-tic field barrier for fluidic particle manipulatio,”Appl. Phys. Lett. , 013702 (2019).[15] M. A. B. Andrade, N. Pérez, and J. C. Adamowski,“Particle manipulation by a non-resonant acoustic levitator,”Appl. Phys. Lett. , 014101 (2015).[16] M Barmatz and P Collas, “Acoustic radiation potential on asphere in plane, cylindrical, and spherical standing wave fields,”J. Acoust. Soc. Am. , 928–945 (1985).[17] B. Raeymaekers, C. Pantea, and D. N. Sinha, “Manipula-tion of diamond nanoparticles using bulk acoustic waves,”J. Appl. Phys. , 014317 (2011).[18] L. D. Landau and E. M. Lifshiftz, Fluid Mechanics (PergamonPress, 1959).[19] Timothy Jones,
Mathieu equation and the ideal RF-Paul trap (Drexel University, 2006). [20] Wolfgang Paul, “Electromagnetic traps for charged and neutralparticles,” Rev. Mod. Phys. , 531–540 (1990).[21] D. Y. Hsieh, “Variational method and mathieu equation,” J.Math. Phys. , 1147 (1977).[22] D. Y. Hsieh, “On Mathieu equation with damping,”J. Math. Phys. , 722 (1980).[23] Adriaan Daniël Fokker, “Die mittlere Energierotierender elektrischer Dipole im Strahlungsfeld,”Ann. Phys. , 810–820 (1914).[24] Max Planck, “Über einen Satz der statistischen Dynamik undseine Erweiterung in der Quantentheorie,” Sitzungsber. König.Preuß. Akad. Wiss. Berlin , 324–341 (1917).[25] L. S. Ornstein, “On Brownian motion,”Proc. Acad. Amst. , 96 (1919).[26] M. Borland, “elegant: A flexible SDDS-compliant code for accelerator simulation,”Tech. Rep. LS-287 (Advanced Photon Source, 2000).[27] Markus Arndt and Klaus Hornberger, “Testingthe limits of quantum mechanical superpositions,”Nature Phys.10