Experimental studies of helium droplets

Quantum solvation dynamics of HCN in a helium-4 droplet

Ultracold nanodroplets of helium-4, containing several thousands of He atoms, offer considerable promise as microscopic cryogenic chambers. Potential applications include the creation of tailor-made chemical or biomolecular complexes and studies of superfluidity in nanoscale systems. Recent experiments have succeeded in interrogating droplets of quantum solvent which consist of as few as 1-20 helium-4 atoms and which contain a single solute molecule. This allows the transition from a floppy, but essentially molecular, complex to a dissolved molecule to be followed and, surprisingly, the transition is found to occur quite rapidly, in some cases for as few as N = 7-20 solvent atoms. For example, in experiments on helium-4 droplets seeded with CO molecules [Tang and McKellar, J. Chem. Phys. 119, 754 (2003)], two series of transitions are observed which correlate with the a-type (Delta K = 0) and b-type (Delta K = +/-1) lines of the binary complex, CO-He (K is the quantum number associated with the projection of the total angular momentum onto the vector connecting the atom and the molecular center of mass). The a-type series, which evolves from the end-over-end rotational motion of the CO-He binary complex, saturates to the nanodroplet limit for as few as 10-15 helium-4 atoms, i.e., the effective moment of inertia of the molecule converges to its asymptotic (solvated) value quite rapidly. In contrast, the b-type series, which evolves from the free-molecule rotational mode, disappears altogether for N approximately 7 atoms. Similar behavior is observed in recent computational studies of HCN(4He)N droplets [Paolini et al., J. Chem. Phys. 123, 114306 (2005)]. In this article the quantum solvation of HCN in small helium-4 droplets is studied using a new fixed-node diffusion Monte Carlo (DMC) procedure. In this approach a Born-Oppenheimer-type separation of radial and angular motions is introduced as a means of computing nodal surfaces of the many-body wave functions which are required in the fixed-node DMC method. Excited rotational energies are calculated for HCN(4He)N droplets with N = 1-20: the adiabatic node approach also allows concrete physical mechanisms to be proposed for the predicted disappearance of the b-type series as well as the rapid convergence of the a-type series to the nanodroplet limit with increasing N. The behavior of the a-type series is traced directly to the mechanics of angular momentum coupling-and decoupling-between identical bosons and the molecular rotor. For very small values of N there exists significant angular momentum coupling between the molecule and the helium atoms: at N approximately 10 solvation appears to be complete as evidenced by significant decoupling of the molecule and solvent angular momenta. The vanishing of the b-type series is predicted to be a result of increasing He-He repulsion as the number of solvent atoms increases.

Electron bubbles in helium clusters. I. Structure and energetics

In this paper we present a theoretical study of the structure, energetics, potential energy surfaces, and energetic stability of excess electron bubbles in ((4)He)(N) (N=6500-10(6)) clusters. The subsystem of the helium atoms was treated by the density functional method. The density profile was specified by a void (i.e., an empty bubble) at the cluster center, a rising profile towards a constant interior value (described by a power exponential), and a decreasing profile near the cluster surface (described in terms of a Gudermannian function). The cluster surface density profile width (approximately 6 A) weakly depends on the bubble radius R(b), while the interior surface profile widths (approximately 4-8 A) increase with increasing R(b). The cluster deformation energy E(d) accompanying the bubble formation originates from the bubble surface energy, the exterior cluster surface energy change, and the energy increase due to intracluster density changes, with the latter term providing the dominant contribution for N=6500-2 x 10(5). The excess electron energy E(e) was calculated at a fixed nuclear configuration using a pseudopotential method, with an effective (nonlocal) potential, which incorporates repulsion and polarization effects. Concurrently, the energy V(0) of the quasi-free-electron within the deformed cluster was calculated. The total electron bubble energies E(t)=E(e)+E(d), which represent the energetic configurational diagrams of E(t) vs R(b) (at fixed N), provide the equilibrium bubble radii R(b) (c) and the corresponding total equilibrium energies E(t) (e), with E(t) (e)(R(e)) decreasing (increasing) with increasing N (i.e., at N=6500, R(e)=13.5 A and E(t) (e)=0.86 eV, while at N=1.8 x 10(5), R(e)=16.6 A and E(t) (e)=0.39 eV). The cluster size dependence of the energy gap (V(0)-E(t) (e)) allows for the estimate of the minimal ((4)He)(N) cluster size of N approximately 5200 for which the electron bubble is energetically stable.

Electron bubbles in helium clusters. II. Probing superfluidity

In this paper we present calculations of electron tunneling times from the ground electronic state of excess electron bubbles in ((4)He)(N) clusters (N=6500-10(7), cluster radius R=41.5-478 A), where the equilibrium bubble radius varies in the range R(b)=13.5-17.0 A. For the bubble center located at a radial distance d from the cluster surface, the tunneling transition probability was expressed as A(0)phi(d,R)exp(-betad), where beta approximately 1 A(-1) is the exponential parameter, A(0) is the preexponential factor for the bubble located at the cluster center, and phi(d,R) is a correction factor which accounts for cluster curvature effects. Electron tunneling dynamics is grossly affected by the distinct mode of motion of the electron bubble in the image potential within the cluster, which is dissipative (i.e., tau(D)<tau(0)) in normal fluid ((4)He)(N) and ((3)He)(N) clusters, while it is undamped (i.e., tau(D)>>tau(0)) in superfluid ((4)He)(N) clusters, where tau(D) is the bubble motional damping time (tau(D) approximately 4 x 10(-12) s for normal fluid clusters and tau(D) approximately 10 s for superfluid clusters), while tau(0) approximately 10(-9)-10(-10) s is the bubble oscillatory time. Exceedingly long tunneling lifetimes, which cannot be experimentally observed, are manifested from bubbles damped to the center of the normal fluid cluster, while for superfluid clusters electron tunneling occurs from bubbles located in the vicinity of the initial distance d near the cluster boundary. Model calculations of the cluster size dependence of the electron tunneling time (for a fixed value of d=38-39 A), with lifetimes increasing in the range of 10(-3)-0.3 s for N=10(4)-10(7), account well for the experimental data [M. Farnik and J. P. Toennies, J. Chem. Phys. 118, 4176 (2003)], manifesting cluster curvature effects on electron tunneling dynamics. The minimal cluster size for the dynamic stability of the bubble was estimated to be N=3800, which represents the threshold cluster size for which the excess electron bubble in ((4)He)(N) (-) clusters is amenable to experimental observation.

Measurement of a long electronic spin relaxation time of cesium atoms in superfluid helium

The longitudinal electronic spin relaxation time of Cs atoms optically polarized in superfluid helium (He II, 1.5 K) has been measured with special care to cope with a serious decrease in the number of Cs atoms in the observation region. This decrease, mainly caused by helium convection in introducing the atoms into He II by laser sputtering, was significantly reduced using a new atom implantation method. Combined with a careful correction for the number of atoms, we have determined the relaxation time to be 2.24(19) s or longer, roughly twice as long as that in solid He.

Magic numbers, excitation levels, and other properties of small neutral He4 clusters (N⩽50)

The ground-state energies and the radial and pair distribution functions of neutral 4He clusters are systematically calculated by the diffusion Monte Carlo method in steps of one 4He atom from 3 to 50 atoms. In addition the chemical potential and the low-lying excitation levels of each cluster are determined with high precision. These calculations reveal that the "magic numbers" observed in experimental 4He cluster size distributions, measured for free jet gas expansions by nondestructive matter-wave diffraction, are not caused by enhanced stabilities. Instead they are explained in terms of an enhanced growth due to sharp peaks in the equilibrium concentrations in the early part of the expansion. These peaks appear at cluster sizes which can just accommodate one more additional stable excitation. The good agreement with experiment provides not only experimental confirmation of the energy level and the chemical potential calculations, but also evidence for a new mechanism which can lead to magic numbers in cluster size distributions. By accounting for the falloff of the radial density distributions at the surface and a size-dependent surface tension, the energy levels are demonstrated to be consistent with a modified Rayleigh model of surface excitations. The compressibility coefficient of these small clusters is found to be one order of magnitude smaller than the bulk compressibility.

Electron-impact ionization of helium clusters close to the threshold: Appearance energies

We have investigated the ionization threshold behavior of small helium cluster ions (cluster size n=2-10) formed via electron-impact ionization of neutral helium droplets and derive appearance energies for mass-selected cluster ions using a nonlinear least-square-fitting procedure. Moreover, we report magic numbers in the mass spectrum observed at the electron energy of 70 eV. The apparatus used for the present measurements is a hemispherical electron monochromator combined with a quadrupole mass spectrometer. Our experiment demonstrates that helium clusters are not only exclusively formed via direct ionization above the atomic ionization potential but also indirectly via autoionizing Rydberg states. The present results are compared with previous electron-impact and photoionization results.

Isotopic replacement in ionic systems: The He2+4+He3→He3He+4+He4 reaction

Full quantum dynamics calculations have been carried out for the ionic reaction (4)He(2) (+)+(3)He and state-to-state reactive probabilities have been obtained using both time-dependent and time-independent approaches. An accurate ab initio potential-energy surface has been employed for the present quantum dynamics and the two sets of results are shown to be in agreement with each other. The results for zero total angular momentum suggest a marked presence of atom exchange (isotopic replacement) reaction with probabilities as high as 60%. The reaction probabilities are only weakly dependent on the initial vibrational state of the reactants, while they are slightly more sensitive to the degree of rotational excitation. A brief discussion of the results for selected higher total angular momentum values is also presented, while the l-shifting approximation [S. K. Gray et al., Phys. Chem. Chem. Phys. 1, 1141 (1999)] has been used to provide estimates of the total reaction rates for the title process. Such rates are found to be large enough to possibly become experimentally accessible.

Fragmentation dynamics of ionized neon clusters (Nen,n=3–14) embedded in helium nanodroplets

We report a theoretical study of the nonadiabatic fragmentation dynamics of ionized neon clusters embedded in helium nanodroplets for cluster sizes up to n=14 atoms. The dynamics of the neon atoms is modeled using the molecular dynamics with quantum transitions method of Tully [J. Chem. Phys. 93, 1061 (1990)] with the nuclei treated classically and transitions between electronic states quantum mechanically. The potential-energy surfaces are derived from a diatomics-in-molecules model to which induced dipole-induced dipole interactions are added. The effect of the spin-orbit interaction is also discussed. The helium environment is modeled by a friction force acting on charged atoms whose speed exceeds the critical Landau velocity. The dependence of the fragment size distribution on the friction strength and on the initial nanodroplet size is investigated. By comparing with the available experimental data obtained for Ne3+ and Ne4+, a reasonable value for the friction coefficient, the only parameter of the model, is deduced. This value is then used to predict the effect of the helium environment on the dissociation dynamics of larger neon clusters, n=5-14. The results show stabilization of larger fragments than in the gas phase, but fragmentation is not completely caged. In addition, two types of dynamics are characterized for Ne4+: fast and explosive, therefore leaving no time for friction to cool down the process when dynamics starts on one of the highest electronic states, and slower, therefore leading to some stabilization by helium when it starts on one of the lowest electronic states.

Path integral Monte Carlo simulation of the absorption spectra of an Al atom embedded in helium

We use a multilevel path integral Monte-Carlo (PIMC) method to simulate the arrangement of He atoms around a single Al atom doped in a He cluster. High-level ab initio Al-He pair potentials and a Balling and Wright pairwise Hamiltonian model are used to describe the full potential and the electronic asymmetry arising from the open-shell character of the Al atom in its ground and excited electronic states. Our calculations show that the doping of the Al 3p electron strongly influences the He packing. The results of the PIMC simulation are used to predict the electronic excitation spectrum of an Al atom embedded in He clusters. With inclusion of tail corrections for the ground and excited states potentials, the calculated 3d<--3p spectrum agrees reasonably well with the experimental spectrum. The blueshift of the calculated spectrum associated with the 4s<--3p transition of solvated Al is about 25 nm (2000 cm-1) larger than seen in experiments on Al embedded in bulk liquid He. We predict that the spectrum associated with the 4p<--3p transition will be blueshifted by approximately 7000 cm-1 (nearly 1 eV).

Infrared spectra of seeded hydrogen clusters: (para-H2)N–N2O and (ortho-H2)N–N2O, N=2–13

High-resolution infrared spectra of clusters containing para-H2 and/or ortho-H2 and a single nitrous oxide molecule are studied in the 2225-cm(-1) region of the upsilon1 fundamental band of N2O. The clusters are formed in pulsed supersonic jet expansions from a cooled nozzle and probed using a tunable infrared diode laser spectrometer. The simple symmetric rotor-type spectra generally show no resolved K structure, with prominent Q-branch features for ortho-H2 but not para-H2 clusters. The observed vibrational shifts and rotational constants are reported. There is no obvious indication of superfluid effects for para-H2 clusters up to N=13. Sharp transitions due to even larger clusters are observed, but no definite assignments are possible. Mixed (para-H2)N-(ortho-H2)M-N2O cluster line positions can be well predicted by linear interpolation between the corresponding transitions of the pure clusters.

Computer Simulation of Quantum Melting in Hydrogen Clusters

We introduce a new criterion, based on multipole dynamical correlations calculated within reptation quantum Monte Carlo, to discriminate between a melting versus freezing behavior in quantum clusters. This criterion is applied to small clusters of para-hydrogen molecules (both pristine and doped with a CO chromophore), for cluster sizes of around twelve molecules. This is a magic size at which para-hydrogen clusters display an icosahedral structure and a large stability. Despite the similar geometric structure of CO@(pH2)12 and (pH2)13, the first system has a rigid, crystalline, behavior; the second behaves more like a superfluid (or, possibly, a supersolid).

The spectra of mixed He3–He4 droplets

The diffusion Monte Carlo technique is used to calculate and analyze the excitation spectrum of 3He atoms bound to a cluster of 4He atoms by using a previously determined optimum filling of single-fermion orbits with well-defined orbital angular momentum L, spin S, and parity quantum numbers. The study concentrates on the energies and shapes of the three kinds of states for which the fermionic part of the wave function is a single Slater determinant: maximum L or maximum S states within a given orbit, and fully polarized clusters. The picture that emerges is that of systems with strong shell effects, whose binding and excitation energies are essentially determined by averages over configuration at fixed number of particles and spin, i.e., by the monopole properties of an effective Hamiltonian.

Ionic reactions in He nanodroplets: the [LiHHe]+ complex and its possible energy pathways into products from ab initio calculations

Ab initio calculations at the multiconfiguration self-consistent field level followed by a multireference configuration interaction were carried out along the two possible collinear approaches of the [LiHHe]+ system, while a three-dimensional calculation of the structures of that complex with LiH+ kept at its equilibrium geometry was also completed at the same level of accuracy. The interaction forces of the lowest two electronic states indicate possible reactive behavior, with the first excited potential-energy surface clearly showing a barrierless path to HeH+ product formation. The details of the reactive pathways and their possible bearing on reaction processes, which could occur at the low temperature of a He nanodroplet holding LiH+ as an impurity, are analyzed and discussed.

Microsolvation of LiH+ in helium clusters: many-body effects and additivity models for the interaction forces

The ab initio calculation of the interaction forces between the LiH+ molecular ion, at its equilibrium geometry, and several He atoms is carried out in order to isolate and assess the importance of many-body contributions in the search for realistic energy and geometry data. The full potential energy surface (PES) with a single helium partner is obtained first by using an aug-cc-pVQZ basis set for He and higher quality ones for Li and H. The calculations were performed at the CAS-SCF plus MRCI level for the lowest potential energy surface over a total of 480 grid points of the two intermolecular Jacobi coordinates, whereas the excited state surface has also been examined in order to exclude the presence of any significant nonadiabatic interaction between the two PESs. A numerical fit of the lower surface is presented and the general physical changes of the ionic interaction when going from the lower to the upper of the two potentials are described and discussed. The fairly limited importance of many-body effects for such systems is seen from further ab initio calculations including several He atoms: our results suggest that, at least in the present case, no strong charge migration occurs after He attachment, and therefore, one could realistically model larger clusters by implementing a sum-of-potentials approach via the presently computed PES.

Near-infrared spectroscopy of ethylene and ethylene dimer in superfluid helium droplets

The nu(5)+nu(9) spectra of ethylene, C(2)H(4), and its dimer, solvated in helium nanodroplets, have been recorded in the wavelength region near 1.6 microm. The monomer transitions show homogeneous broadening of approximately 0.5 cm(-1), which is interpreted as due to an upper state vibrational relaxation lifetime of approximately 10 ps. Nearly resonant vibrational energy transfer (nu(5)+nu(9)-->2nu(5)) is proposed as the relaxation pathway. The dimer gives a single unresolved absorption feature located 4 cm(-1) to the red of the monomer band origin. The scaling of moments of inertia upon solvation in helium is 1.18 for the monomer and >2.5 for the dimer. In terms of the adiabatic following approximation, this classifies the monomer as a fast rotor and the dimer as a slow rotor.

Small para-hydrogen clusters doped with carbon monoxide: Quantum Monte Carlo simulations and observed infrared spectra

The structures and rotational dynamics of clusters of a single carbon monoxide molecule solvated in para-hydrogen, (paraH(2))(N)-CO, have been simulated for sizes up to N=17 using the reptation Monte Carlo technique. The calculations indicate the presence of two series of R(0) rotational transitions with J=1<--0 for cold clusters, similar to those predicted and observed in the case of He(N)-CO. Infrared spectra of these clusters have been observed in the region of the C-O stretch ( approximately 2143 cm(-1)) in a pulsed supersonic jet expansion using a tunable diode laser probe. With the help of the calculations, the observed R(0) rotational transitions have been assigned up to N=9 for the b-type series and N=14 for the a-type series. Theory and experiment agree rather well, except that theory tends to overestimate the b-type energies. The (paraH(2))(12)-CO cluster is calculated to be particularly stable and (relatively) rigid, corresponding to completion of the first solvation shell, and it is observed to have the strongest a-type transition.

Controlling the coulomb explosion of silver clusters by femtosecond dual-pulse laser excitation

Silver clusters grown in helium nanodroplets are excited by intense femtosecond laser pulses resulting in the formation of a hot electron plasma far from equilibrium. The ultrafast dynamics is studied by applying optically delayed dual pulses, which allows us to pursue and control the coupling of the laser field to the clusters on a femtosecond time scale. A distinct influence of the optical delay on the ionization efficiency gives strong evidence that a significant contribution of collective dipolar electron motion is present, which is verified by corresponding Vlasov dynamics simulations on a model system. The microscopic approach demonstrates the outstanding role of giant resonances in clusters also in intense laser fields.

Tunable diode laser spectrometer for pulsed supersonic jets: application to weakly-bound complexes and clusters

The design and operation of an apparatus for studying infrared spectra of weakly-bound complexes is described in detail. A pulsed supersonic jet expansion is probed using a tunable Pb-salt diode laser spectrometer operated in a rapid-scan mode. The jet may be fitted with either pinhole or slit shaped nozzles, the former giving lower effective rotational temperatures, and the latter giving sharper spectral lines. Notable features of the apparatus include use of a toroidal multi-pass mirror system to give over 100 passes of the laser through the supersonic jet, use of the normal laser controller for laser sweeping during both setup and data acquisition, and use of a simple semi-automated wavenumber calibration procedure. Performance of the apparatus is illustrated with observed spectra of the van der Waals complex He-OCS, and the seeded helium clusters He(N)-OCS and He(N)-CO.

Infrared spectra of seeded hydrogen clusters: (paraH2)N–OCS, (orthoH2)N–OCS, and (HD)N–OCS, N=2–7

Infrared spectra of hydrogen-carbonyl sulfide clusters containing paraH2, orthoH2, or HD have been studied in the 2060 cm(-1) region of the C-O stretching vibration. The clusters were formed in pulsed supersonic jet expansions and probed using a tunable infrared diode laser spectrometer. Simple symmetric rotor type spectra were observed and assigned for clusters containing up to N = 7 hydrogen molecules. There was no resolved K structure, and Q-branch features were present for orthoH2 and HD but absent for paraH2. These characteristics can be rationalized in terms of near symmetric rotor structures, very low effective rotational temperatures (0.15 to 0.6 K), and nuclear spin statistics. The observed vibrational shifts were compared with those from recent observations on the same clusters embedded in helium nanodroplets. The observed rotational constants for the paraH2 clusters are in good agreement with a recent quantum Monte Carlo simulation. Some mixed clusters were also observed, such as HD-HD-He-OCS and paraH2 - orthoH2 - OCS.

Excited-state relaxation of Ag8 clusters embedded in helium droplets

Neutral silver clusters Ag(N) are grown in ultracold helium nanodroplets. By exploiting a strong absorption resonance recently found for Ag8, first photoelectron spectra of this neutral species are recorded. Variation of the laser photon energy reveals that direct vertical two-photon ionization is hindered by rapid relaxation into the lower edge of a long-living excited state manifold. The analysis of the dynamics gives a precise value of (6.89+/-0.09) eV for the vertical ionization potential of Ag8. The influence of the helium matrix on photoemission is discussed.

Formation and stability of high-spin alkali clusters

Helium nanodroplet isolation has been applied to agglomerate alkali clusters at temperatures of 380 mK. The very weak binding to the surface of the droplets allows a selection of only weakly bound, high-spin states. Here we show that larger clusters of alkali atoms in high-spin states can be formed. The lack of strong bonds from pairing electrons makes these systems nonmetallic, van der Waals-like complexes of metal atoms. We find that sodium and potassium readily form such clusters containing up to 25 atoms. In contrast, this process is suppressed for rubidium and cesium. Apparently, for these heavy alkalis, larger high-spin aggregates are not stable and depolarize spontaneously upon cluster formation.

Potential energy surface, bound states, and rotational inelastic cross sections of the He-CH[sub 4] system: A theoretical investigation

We determined two potential energy surfaces (PES) for the He-CH(4) system by means of MP4 and Valence Bond (VB) calculations. The MP4 potential is similar to the one commonly adopted for this system [U. Buck, K. H. Kohl, A. Kolhase, M. Faubel, and U. Staemmler, Mol. Phys. 55, 1255 (1985)], while the VB PES is slightly more attractive. To evaluate the reliability of these potentials, we investigated the scattering properties by performing close coupling calculations, and concluded that: (i) the available experimental data do not permit the ranking among the PES considered; (ii) some theoretical predictions differ considerably from the experimental data, and these discrepancies cannot entirely be ascribed to the inaccuracy of the ab initio calculations; (iii) the scattering properties at low energy might discriminate between the MP4 and VB potentials.

High resolution infrared spectra of a carbon dioxide molecule solvated with helium atoms

Infrared spectra of He(N)-CO(2) clusters with N up to about 20 have been studied in the region of the CO(2) nu(3) fundamental band ( approximately 2350 cm(-1)) using a tunable diode laser spectrometer and pulsed supersonic jet source with cooled (>-150 degrees C) pinhole or slit nozzles and high backing pressures (<40 atm). Compared to previous studies of He(N)-OCS and -N(2)O clusters, the higher symmetry of CO(2) results in simpler spectra but less information content. Discrete rotation-vibration transitions have been assigned for N=3-17, and their analysis yields the variation of the vibrational band origin and B rotational constant over this size range. The band origin variation is similar to He(N)-OCS, with an initial blueshift up to N=5, followed by a monotonic redshift, consistent with a model where the first five He atoms fill a ring around the equator of the molecule, forcing subsequent He atom density to locate closer to the ends. The B value initially drops as expected for a normal molecule, reaching a minimum for N=5. Its subsequent rise for N=6 to 11 can be interpreted as the transition from a normal (though floppy) molecule to a quantum solvation regime, where the CO(2) molecule starts to rotate separately from the He atoms. For N>13, the B value becomes approximately constant with a value about 17% larger than that measured in much larger helium nanodroplets.

Photoelectron imaging of helium droplets

The photoionization and photoelectron spectroscopy of He nanodroplets (10(4) atoms) has been studied by photoelectron imaging with photon energies from 22.5-24.5 eV. Total electron yield measurements reveal broad features, whose onset is approximately 1.5 eV below the ionization potential of atomic He. The photoelectron spectra are dominated by very low energy electrons, with <E(k)> less than 0.6 meV. These results are attributed to the formation and autoionization of highly vibrationally excited He(*)(n) Rydberg states within the cluster, followed by strong final state interactions between the photoelectron and the droplet.

Bubble formation and decay in 3He and 4He clusters

The energy transfer in 3He and 4He clusters electronically excited by monochromatic synchrotron radiation is investigated by luminescence spectroscopy. Depending on the cluster size and the isotopic constitution, either sharp, broadened, or shifted emission bands of single He molecules are observed. The spectral features show that He molecules emit light either within a bubble inside the cluster or in the vacuum after desorption from the cluster. From the luminescence intensity, the cluster diameter, and the radiative lifetime, an average velocity of approximately 7 m/s of bubbles in 4He clusters could be deduced. In the nonsuperfluid 3He clusters this velocity was observed to be significantly lower.