1. Introduction

In this paper we describe an approach to the solution of the phase problem for two-dimensional crystals, based on the idea that the compact support constraint may be applied along the direction normal to a periodic slab - that is, along the beam direction z. This constraint is then included in the iterative Gerchberg-Saxton-Feinup hybrid-input-output (HiO) algorithm [1]. In addition, we outline plans to trap nanostructures with laser tweezers in an electron microscope for atomic-resolution diffractive imaging experiments, thereby eliminating background scattering and providing the required isolated object and “support” function. We refer to the inversion of single-scattering diffraction patterns from objects, which do not have three-dimensional translational symmetry as Coherent Diffractive Imaging (CDI). When based on the HiO algorithm, the boundary of the object must be approximately known (“support constraint”), together with the sign of the scattering density and the measured diffraction intensities (“Fourier modulus constraint”).

Some recent advances in CDI can be briefly summarized. Soft X-ray transmission patterns from lithographed patterns were first inverted to images using the HiO algorithm in 1999 [2]. More recently, similar patterns at higher resolution have been inverted using support estimates based on the support of the autocorrelation function of a non-periodic object [3,4], and a new algorithm has appeared in which the support function (initially based on the autocorrelation function) is continuously updated and improved during the HiO iterations [5]. In that case, little or no a-priori information regarding the unknown object is needed apart from the diffracted intensities. First applications of the HiO-algorithm to electron diffraction were reported by Weierstall et al. in 2001 [6], and striking images of a double-walled carbon nanotube have recently been obtained at atomic resolution by this method [7].

Several diffraction methods for solving protein structures, which either cannot or are very difficult to crystallize, are currently under development. These proteins include the very important class of cell membrane proteins used for drug delivery. Many of them, while difficult to crystallize, can nevertheless be encouraged to form two-dimensional crystals. The new methods include the proposed use of pulsed X-ray sources [8], which provide diffraction patterns from individual molecules before they are destroyed, together with the proven methods of cryomicroscopy. In cryomicroscopy, either two-dimensional crystals or individual, randomly oriented molecules are embedded in a medium (often amorphous ice) a few hundred Angstroms thick and imaged in a high-resolution electron microscope (TEM). For the crystals, diffraction patterns and corresponding images are obtained over a wide range of orientations. The high-resolution phase-contrast images, (at perhaps 0.3nm resolution) are corrected for lens aberration effects, then used to find, by Fourier Transform, the phases of the diffraction pattern intensities. The redundancy provided by the use of two-dimensional crystals then addresses the radiation-damage problem: If the dose per molecule is less than that needed to destroy it, the dose is insufficient to produce a statistically significant scattering pattern from one molecule. If the same dose is used for scattering from an array of identical molecules, statistically significant data is generated. A common situation with both organic and inorganic films studied by TEM is that diffraction patterns are much easier to obtain experimentally, and with much higher resolution, than the TEM images, particularly when the thin film is inclined at a large angle to the optic axis, as needed for the collection of tomographic data.

In this paper we show, using simulations for a small organic molecule, that the number of such images needed to phase the three-dimensional diffraction data from a two-dimensional crystal can be greatly reduced (from hundreds to a few) by using iterative phasing methods together with the compact support condition applied normal to the boundaries of the slab (z-direction). Two-dimensional crystals have a periodic boundary condition in two dimensions and are non-periodic along a third. The application of a one dimensional compact support condition in combination with direct methods to solve three dimensional surface structures of bulk crystals from X-ray reflection diffraction patterns in surface science has been analyzed theoretically in [9]. We are interested in determining the minimum number of known phases, which must be supplied from electron microscope images, perhaps at limited resolution, to phase the entire three-dimensional reciprocal space when the compact support condition can be applied along z. Since knowledge of phase is a convex constraint [10], these images greatly improve convergence. In this way, by requiring fewer images, the time and effort devoted to data collection and analysis in electron crystallography might be greatly reduced. It is well known that, apart from the special case of disjoint support, the HiO- algorithm does not converge in one dimension, nor have means been found to apply it to three-dimensional crystals other than through the use of non-crystallographic symmetries [11]. The case of a two-dimensional crystal of finite thickness therefore seems worthy of investigation. Questions of the uniqueness of solutions, convergence, error metrics, relationship to steepest-descent optimizers, and an illuminating analysis using the methods of projections onto convex and non-convex sets are all considered elsewhere [12].

The Shannon sampling theorem shows that the continuous distribution of diffracted intensity from a single molecule of width W can be obtained at any angle by interpolation from sampled values of the diffracted intensity if the samples occur at an angular spacing proportional to 1/(2W) (in one dimension). This spacing is the reciprocal of the width 2W of the Fourier Transform of the diffracted intensity, which is the autocorrelation function of the molecule, and so plays the role of the “bandlimit” in Shannon’s theory. All the information contained in the scattered intensity (usually enough to solve the phase problem) can only be extracted if this Shannon sampling is used [13]. Periodic repetition of the molecule on a lattice with spacing W, however, will generate Bragg peaks at twice this angular separation, and hence the Bragg intensities alone (without other a-priori assumptions) do not contain enough information to solve the phase problem. Thus no way has been found to use the HiO algorithm to solve the phase problem for large three-dimensional crystals.

Shannon sampling of the diffracted intensity (at intervals of 1/2W) has been referred to as “oversampling” (by a factor of two), and corresponds to Bragg sampling in the situation in which the molecule is surrounded by a border of material (such as a water jacket or vacuum) which has the effect of doubling the unit cell dimensions. The imposition of “known” zeros in this region in real space during iterations can be thought of as compensating for the lack of a corresponding amount of phase information in reciprocal space, since the result makes the number of Fourier equations just equal to the number of unknown phases. In this sense, Shannon sampling is needed to extract all information, including phases, from the diffracted intensity. The HIO algorithm is then capable of finding the very large number of adjustable parameters (one phase for each diffracted pixel) given the compact support constraint (known region of zero charge density surrounding the molecule). The algorithm is tolerant to noise in simulations, however it has proven difficult to make the isolated, “real” (single scattering) samples with approximately known support, which are needed for experiments [3]. Several approaches to the problem of estimating the support of an object from the support of its autocorrelation function have been suggested [1]. Detailed descriptions of the HIO-algorithm, which iterates between real and reciprocal space, supplying known information in both domains, have been given elsewhere [6]. To monitor the progress of the algorithm, an image space error metric ε_k is calculated during every iteration. This root-mean square (rms) error ε_k is the amount by which the reconstructed potential violates the image-space constraints, i.e. it measures the deviation of the current estimate of the potential from zero outside the slab [6].

Figure 1 schematically shows the application of the support condition normal to a monolayer crystal along z. For a thin crystal of thickness t (we assume the crystal to be one molecule thick), the Fourier domain consists of reciprocal-space rods (relrods) normal to the crystal. The extent of the autocorrelation function along z in real-space is 2t, which defines the Shannon sampling interval of Δθ=λ/2t in reciprocal space as shown. (Our simulations are based on an interval λ/3t).

 

Fig. 1. Geometry of “oversampling” for a two-dimensional crystalline slab. The compact support constraint is applied in real space normal to the molecular monolayer. In reciprocal space the intensities along the relrods are assumed known from a diffraction pattern tilt series. Known phases are supplied on a plane normal to the relrods and two planes tilted by 45 degrees. The HIO-algorithm then reconstructs the phases along the relrods.

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2. Small molecules at atomic resolution: Tetracyanoethylene

We report here simulations for the tetracyanoethylene (TCNE) small molecule arranged here as a two-dimensional crystal. Figure 2 shows the structure of the TCNE crystalline monolayer used in the simulations. TCNE, C₆N₄, is body-centered cubic, with spacegroup Im3 and cell constant 9.736 Å [14]. The three inequivalent molecules per cell are shown in Fig. 1. A center of symmetry at (1/4,1/4,1/4) generates three more molecules that are arranged in a mutually perpendicular fashion.

 

Fig. 2. Arrangement of molecules in the cubic unit cell of a tetracyanoethylene (TCNE) crystal. Additional molecules are generated by a center of inversion at (1/4, 1/4, 1/4). The center of inversion for the molecule lies at the midpoint of the cell axes

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Using a Gaussian expansion for the relevant relativistic Hartree-Fock electron scattering factors [15] we have simulated the kinematic diffraction pattern of amplitudes for a single layer of these molecules. We assume that the moduli of the Fourier coefficients of the electrostatic potential can be extracted from experimental diffraction patterns after correction for effects due to film bending, curvature of the Ewald sphere and other artifacts. A temperature factor exp(-B s ²) with B=0.3 and s=1/d_hkl was assumed. Since the molecule is centrosymmetric, the structure factors are real, and the phase problem consists of a sign ambiguity. Application of the HiO algorithm to solve this structure requires “oversampling” of the data along the relrod direction z normal to the film, which are separated laterally by multiples of the Bragg angle. The Bragg scattering angles are used for lateral dimensions normal to z.

To summarize, the magnitudes of the Fourier coefficients of potential for a tetracyanoethylene monolayer were used as input to the HiO algorithm with the addition of a compact support condition normal to the slab. In practice a computational supercell was used whose dimensions were equal to the crystal monolayer unit cell dimension laterally (real space sampling interval: 0.03nm, super cell size in z and y: 30×30 pixels), but three times this dimension measured along z, normal to the monolayer (supercell size in z: 90 pixels). This corresponds to an oversampling factor N=3, which then fixes the points at which samples are taken along the relrods. While oversampling by 2 satisfies the Shannon requirement, it has been suggested in X-ray HiO work [2] that better convergence is obtained at somewhat higher factors.

If no additional phases were provided a-priori, it was found that the HIO algorithm with compact support along z did not converge. We then assumed that phases could be provided from high-resolution images recorded in various orientations. If known phases where provided on three different planes, the algorithm has been found to converge after about 100 iterations (in the sense that a small value of ε_k is soon obtained and a recognizable charge-density obtained, as in Fig. 4). Known phases were provided on the plane in reciprocal space normal to the direction of the incident beam passing through the origin, together with planes tilted at 45 degrees about the x and y axis. Figure 4 shows an isopotential view of a contour of the electrostatic potential for the TCNE molecule. The potential was reconstructed after 70 iterations with the 30×30×90 voxel data set with the feedback parameter β=0.8. The simulated diffracted intensities (to 0.35 Å) were combined with known phases from three simulated images as described above, also to 0.35 Å resolution. The support mask sets the object function to zero for slices at z values equal to or less than 28 and greater than or equal to 62, corresponding to the vacuum before and after the sample. The algorithm converges to values of ε_κ of about 0.047 after only 70 iterations, independent of the starting phases. (In terms of the error metrics defined in the next section, after 105 iterations we find a correlation coefficient between the estimated and true charge density of CC=0.995, ε_κ=0.047). These calculations do not simulate the effects of noise.

Since the rms-error ε_k in the HIO-algorithm is not always a measure of the quality of the reconstructed image [1], the correlation coefficient (CC) between the true and estimated potential was calculated after each iteration. CC is equal to the value of the normalized cross-correlation function at the origin, or [16]

Here ρ_t(r) and ρ_e(r) are true and estimated electron charge densities respectively. F_h are the measured structure factors used in the HiO algorithm, which adjusts only phases. The value of CC lies in the range CC=-1 (anticorrelated) to CC=1 (perfect agreement). Figure 3 shows CC and the rms-error ε_k against the iteration number for TCNE with phases supplied on one and three planes. The algorithm didn’t converge for phases known only on one plane. For phases known on three planes, the algorithm converged to values of ε_κ of about 0.047 after only 70 iterations, independent of the starting phases. The correlation coefficient CC between the estimated and true charge density was then CC=0.995. These calculations do not simulate the effects of noise.

 

Fig. 3. Correlation coefficient CC (solid line) between the estimated and true charge density and the HIO rms-error ε_k (dashed line) against the iteration number for TCNE with known phases supplied on one (green) and three planes (red). The algorithm didn’t converge for phases known only on one plane (green). For phases known on three planes, the algorithm converged to values of ε_κ of 0.047 after 70 iterations. The correlation coefficient CC was then CC=0.995, which means almost perfect agreement between estimated and true charge density. The algorithm consists of cycles of 25 HIO iterations followed by 10 error-reduction iterations, which explains the periodic structure in the rms-error. When the HIO-iterations start after 10 error reduction iterations (at iteration number 35 and 70) the rms-error gets worse but improves soon after. For details see [6].

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Fig. 4. (300kB) Movie of HIO iterations. Isopotential view of the electrostatic potential of TCNE as reconstructed by the HiO algorithm from simulated diffracted intensities (to 0.35 Å) combined with 3 images (to 0.35 Angstroms). The computational supercell is shown, the size in the z (horizontal) direction is three times the thickness of the monolayer, providing for the compact support. The size in the x and y direction is the size of one unit cell (0.97nm). 70 iterations were needed for the 30×30×90 voxel data set. The movie is assembled from the resulting images after every 5^th iteration.

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3. Electron diffraction with laser tweezers

A crucial experimental problem with all diffractive imaging experiments is the need to provide a supporting membrane for the object, which does not generate background scattering. The object must be isolated, and thus, in the case of electron diffraction, where a very strong beam-sample interaction exists, should preferably be surrounded by a border of vacuum. This might be achieved if nano-particle samples were to be supported by laser tweezers. Early experiments on the levitation of small dielectric particles [17] have demonstrated that these may be supported in vacuum at the focus of a laser beam of suitable form. Subsequent work by these authors has shown that particles as small as 10 nm can be trapped by a highly focused laser beam [18], which formed the basis of the entire laser tweezers industry.

We are therefore constructing a side-entry TEM holder to test this idea. This device will be a modified single-tilt holder, with an optical fiber running along its length fed by a solid-state laser. A large numerical aperture lens will be coupled to the end of the fiber, focused at the optic axis of the electron microscope. At this point it should be possible to trap and examine small particles, such as magnesium oxide cubes of nanometer dimensions.

The stability of nanometer size particles in the optical trap, in the presence of perturbing forces, may be estimated using the analysis given by Ashkin et al. [18]. For a magnesium oxide cube of edge length 40 nm the force due to gravity is attonewtons, while the trapping force is of the order of piconewtons for a laser power of tens of milliwatts. The laser diode that we will use has a maximum power of 70 mW. The force due to momentum transfer from a 10pA, 100keV electron beam is also a few attonewtons in a strong two beam diffraction condition from magnesium oxide. The total recoil momentum from the secondary electron emission process will be negligible since each individual recoil direction is nearly random. Charging of the particle due to secondary emission is more problematic. One can suppose that a magnesium oxide particle will charge positively to the second critical voltage (quoted variously from 3.5 to greater than 5 kV) and be stable in the electron beam, since the surrounding potential would be roughly cylindrically symmetric and near ground. Alternatively a low energy electron or ion flood gun could be used to neutralize the particle charge.

4. Discussion

Extensions of these simulations to proteins are also under way [19]. In this paper we have shown that repeated application of the boundary condition for a thin slab in an iterative algorithm can greatly assist solution of the phase problem for a two-dimensional crystal. Known phases are then only needed from images recorded over a small range of tilts in order to phase the entire reciprocal space volume, thus avoiding the need to record images at high tilt angles. The main advantage of the method presented here is the reduction in time and effort needed to complete a three-dimensional structural analysis, since image recording - especially at high tilts - is much more difficult than the recording of diffraction patterns. Although we need to sample the reciprocal lattice rods more finely, we only need to include more of the readily obtainable diffraction data. Thus there is some tradeoff in increasing the degree of oversampling to improve convergence versus increasing the number of diffraction patterns required. With 2× oversampling, in fact a similar number of data sets (images plus diffraction patterns) would be required as for the normal approach [20, 21], but these would be all diffraction patterns.

It was first pointed out by Pauling for the case of bixbyite that two different crystal structures (which are not enantiomorphs) may have the same diffraction pattern intensities (see Ref. [22] for a review). One class of these “homometric” ambiguous structures are those defined by the convolution relations given in Ref. [23]. We have performed HiO simulations for these rare ambiguous structures, and find that the HiO algorithms can only distinguish them if sufficiently tight support is provided.

These simulations have made no use of the known space group symmetries of the infinite structure, which may be reduced to those of a slab for a monolayer. Thus an optimized program might be written based solely on inequivalent reflections, in which the voxel number is greatly reduced and the speed of convergence increased. The power of the support condition may also be applied laterally if some pixels of known “zero” potential can be applied within the unit cell. This method, akin to density modification or solvent flattening, would rapidly improve convergence.

A modest “phase extension” effect was found in the simulation for tetracyanoethylene, but the use of HiO-algorithm to improve the resolution of images using higher-resolution diffraction data has not been fully explored here.