Ultrashort electron pulses for diffraction, crystallography and microscopy: theoretical and experimental resolutions

Transcription

1 PAPER Physical Chemistry Chemical Physics Ultrashort electron ulses for diffraction, crystallograhy and microscoy: theoretical and exerimental resolutions Andreas Gahlmann, Sang Tae Park and Ahmed H. Zewail* Received 6th February 2008, Acceted 6th March 2008 First ublished as an Advance Article on the web 31st March 2008 DOI: /b802136h Pulsed electron beams allow for the direct atomic-scale observation of structures with femtosecond to icosecond temoral resolution in a variety of fields ranging from materials science to chemistry and biology, and from the condensed hase to the gas hase. Motivated by recent develoments in ultrafast electron diffraction and imaging techniques, we resent here a comrehensive account of the fundamental rocesses involved in electron ulse roagation, and make comarisons with exerimental results. The electron ulse, as an ensemble of charged articles, travels under the influence of the sace charge effect and the sread of the momenta among its electrons. The shae and size, as well as the trajectories of the individual electrons, may be altered. The resulting imlications on the satiotemoral resolution caabilities are discussed both for the N-electron ulse and for single-electron coherent ackets introduced for microscoy without sace charge. I. Introduction In the investigation of comlex systems ranging from biology to chemistry to materials science, it is beneficial to obtain structural information as a function of time. To achieve this goal, our laboratory has develoed 1 the ultrafast techniques of electron diffraction (UED), 2 electron crystallograhy (UEC) 3,4 and electron microscoy (UEM); 5 for more details of the historical develoment see ref. 1 and references therein. In striving to identify the relevant degrees of freedom of the structural dynamics, the exerimental tools for investigating fundamental hysical, chemical and biological rocesses need to feature ever-imroving satiotemoral resolution. Using electrons as a robe, the resulting high sensitivity allows for the use of ultrashort ulsed beams with unrecedented time resolution, ranging from femtoseconds to icoseconds. The temoral resolution is mainly determined by the longitudinal extent of the electron ulse, while the satial resolution limits for both diffraction and imaging are determined by the same requirements that aly to continuous-wave beams. To establish and ossibly imrove uon the resolution limits of ultrafast electron diffraction and electron imaging instruments, detailed knowledge of the temoral evolution of electron ulses becomes crucial. Here, we resent a comrehensive theoretical study of the relevant ulse broadening mechanisms and investigate their effect on the satiotemoral resolution in electron diffraction, crystallograhy, and imaging. The geometrical factors caused by the velocity mismatch between the otical and electron ulses in a crossed beam arrangement are treated elsewhere 6 and are not considered here. As discussed below, a tilted ulse geometry circumvents this roblem in UEC. Physical Biology Center for Ultrafast Science and Technology, Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, CA zewail@caltech.edu The electron ulse trajectories are concetually similar in all of the above-mentioned instruments. The ultrashort electron ulse is formed by illuminating a hotocathode material with a ulsed laser beam, thereby generating free electrons by the hotoelectric effect. The free electron acket is then accelerated through an electric field and subsequently shaed using an instrument secific combination of inholes, electrostatic and magnetostatic lenses, as shown schematically in Fig. 1. However, the number of electrons er ulse varies considerably across our instruments: in gas-hase electron diffraction (UED), where the secimen s molecular density is low, the electron ulse contains u to 10 6 electrons in order to maximize the signal intensity, while in electron crystallograhy (UEC), B10 3 electrons er ulse results in sufficient scattering events (see Aendix A). Finally, in electron microscoy (UEM), single-electron ulses are emloyed at an increased reetition rate to build u an image or diffraction attern in time. Several factors come into lay during the electron ulse trajectory and these factors determine the bunch shae at the secimen in the interaction region. First, the electrons are generated by a laser ulse, which itself has a temoral and satial extent; the extent defines the initial shae of the electron acket. Second, the electrons are ejected from the hotocathode with a range of momenta defined by the distribution in magnitude and direction. Third, at sufficiently high charge densities, air-wise Coulomb reulsions between electrons become significant and such reulsions may alter the velocities of individual electrons. The latter effect is not oerational in the single-electron mode. Simulating electron ackets realistically along the entire ath is comlicated due to the resence of instrument secific electron otical arrangements. While the electron microscoe features a myriad of magnetostatic lenses, electrostatic lenses, deflection coils, stigmator coils and inholes, the column of our UED or UEC instruments, containing just a single 2894 Phys.Chem.Chem.Phys., 2008, 10, This journal is c the Owner Societies 2008

2 II. Theory A. Definition of width and length of acket Fig. 1 Schematic of the otical column in UED/UEC and UEM. Electrons are generated by the hotoelectric effect at the cathode (C) with the given rofile, accelerated between a single electrode air, radially focused by a solenoid coil (M). The electron ulse evolution is monitored, from the source until they reach the detector (D). In UEM, the ulses are shaed using lens systems (L1, L2, and L3), rather than simle solenoid coils. magnetostatic lens, seems fairly simle by comarison. Theoretical studies to date 7 11 have treated electron ulses for diffraction and imaging rimarily in the absence of focusing fields, thereby relying on several assumtions to match exerimental data, which could be violated in reality. This aer is organized as follows: First we will consider the roagation of electron ulses in the absence of Coulomb reulsions to determine the magnitude of ulse broadening mechanisms, other than the sace charge effect. Second, the electron ulses in the resence of sace charge will be modeled with tyical UED or UEC arameters using two aroaches: a mean field model 7,8,10 and an N-body Monte Carlo simulation. 12 The mean field model will be exanded to allow for the incororation of ulse shaing fields. It is shown that the Monte Carlo simulations give more accurate ulse shaes and, erhas more imortantly, deliver the comlete hase sace information of the ensemble of electrons. Together these two aroaches rovide a valuable illustration of the hysical rocesses of electron electron interactions within the ulse and their subsequent imlications for electron secimen interactions in diffraction and imaging. We conclude with a discussion about the beam coherence roerties and aly these concets to maximize the satiotemoral caabilities of our newly-develoed diffraction instrument, UED4. Comarisons with the exerimental resolutions achieved are also made (see Fig. 2). Macroscoically, the electron acket is defined by its longitudinal and lateral satial rofiles and their evolution in time, as it travels from the hotocathode to the interaction region and, finally, to the detector. Tyically, these distributions can be assumed to be cylindrically uniform, ellisoidal, Gaussian or their combinations. 7 When comaring different shaes and distributions of electron ackets, a natural measure for the satial extent has to be defined. While the maximum extent would be an obvious choice for a uniform distribution, the full-width-half-maximum (FWHM) is often used for a Gaussian distribution and for exerimental measurements; in few cases, the width at 10% height and/or the standard deviation, s, is also used. For the uniform cylindrical and/or ellisoidal distribution, the measurement of the diameter often imlies a rojection of the 2D/3D density onto the resective axis, resulting usually in a Gaussian-tye function. Furthermore, real electron ackets do not necessarily maintain their initial shaes, during exansion, with the noted excetion of ellisoidal ackets. 13 For quantitative comarison of the results obtained from the mean field theory, the numerical simulation, and exeriment, we choose the standard deviation as the universal metric of size. Two ackets are deemed equivalent when they have the same standard deviations regardless of the shae of the distribution. Only if the actual distribution shae is known or assumed, then the standard deviation can be related to more common measures of the satial extent of the ulse, such as the FWHM (see Aendix B for conversions). In the mean field theory, for examle, in order to simulate a 110 fs FWHM Gaussian ulse (s = 47 fs), we would choose a uniform length of 162 fs ð¼ ffiffiffiffiffi fsþ. Alternatively, Miller and coworkers, 11 instead of converting the measure of size, scaled the number of electrons in the ackets to match the results of the mean field theory and their N-body simulation, since the FWHM of a Gaussian distribution contains only 76% of the total electrons. B. Initial energy sread In the hotoelectric rocess, free electrons are generated with a distribution of kinetic energies. In order to evaluate the magnitude of the ulse broadening it is instructive to treat the sace charge effect searately from the broadening due to the initial energy sread. To this end, the relativistic equation of motion has been solved for a single electron having an uncertainty in its momentum. Longitudinal broadening, Dt KE, will result from an initial sread in the electron kinetic qenergy, ffiffiffiffiffi m DE i, or the corresonding momentum sread, D i ¼ 0 2E i DE i, which occurs during the electron hotoemission event (see Aendix C): Dt KE ¼ d ev 1 v i l v i v f m 0 g 3 f v 3 D i ð1þ f where e is the electron charge, m 0 is the electron rest mass, d is the acceleration ga between cathode and anode, l is the distance of field-free roagation, V is the acceleration This journal is c the Owner Societies 2008 Phys.Chem.Chem.Phys., 2008, 10,

3 Fig. 2 Measured resolutions for UEC and UED and exerimental transients obtained by UEC and UEM1. (a) Streaked electron ulses on the CCD (charge-couled device) detector together with the calculated ulse lengths. (b) Measured electron ulse widths as a function of the number of electrons. The blue curve (UED3) shows more than an order-of-magnitude imrovement in the electron gun erformance in comarison to the red curve (UED2). The number of electrons for the UED3 measurement in ref. 20 was given as density (electrons mm 2 ). For the data shown here, the original streak images have been reanalyzed and they are now given in terms of the absolute number of electrons. The lines are drawn as best fits, but the theoretical curves are given in Fig. 6. (c) Ultrafast dynamics of structural hase transition in vanadium dioxide. Intensity change of the (606) Bragg sot with time. A decay with a time constant t 1 of 307 fs was reorted in ref. 31. Here the data was deconvoluted (electron ulse width of 344 fs) and we obtained t 1 = s. (d) Temoral evolution of the structural order arameter. The order arameter is defined as the integrated intensity of the diffraction eak for different temoral frames. Adoted from ref. 3, 20, 31, and 38. voltage, v i and v f are the mean initial and final electron velocities, g f is the relativistic Lorentz factor (see Aendix C) at the velocity v f. The first term reresents the broadening of the electron ulse in the acceleration ga, the dominant contribution, while the second term reresents broadening in the field-free drift region. We note that the second term corresonds to a acket simly sreading by l/(v f Dv f ) in the drift region. Since v i { v f, the exression can be aroximated to yield Dt KE d ev D i ¼ d ev rffiffiffiffiffiffi m 0 DEi ffiffiffiffi 2 E i ð2þ giving a result equivalent to that of eqn (4) in ref. 14. It should be noted that, under this aroximation, the temoral sread is solely determined by the energy sread, DE i, relative to the ffiffiffiffiffi square root of the mean energy, E i, and the otential V across the distance d. C. Charge density effect 1. Mean field theory (MF). In the mean field theory, which has been widely used in the literature, 7 11 the electron ulse is most commonly aroximated as a cylindrical slab of radius R and length L. Due to the sace charge effect, this slab of continuous charge density extends both in the longitudinal 2896 Phys.Chem.Chem.Phys., 2008, 10, This journal is c the Owner Societies 2008

4 and transverse direction, while always maintaining the shae of a cylinder. The evolution of the satial ulse length, L, due to sace charge reulsion is given by (see Aendix D for derivation): 1 d 2 L 2 dt 2 ¼ a X ¼ Ne2 2 2m 0 e 0 R 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ L R þ 1 þ L 2 R where a X is the axial acceleration, N is the number of electrons in the ulse, and e 0 is the vacuum ermittivity. The time evolution of the radius, R, of a freely exanding, thin disk (R c L) is aroximated by 8 d 2 R dt 2 ¼ a R Ne2 m 0 e 0 R 2 ð4þ where a R is the radial acceleration. The temoral sread, Dt SC, is then evaluated from the length of acket and its axial velocity, Dt SC ¼ L ð5þ v f Here, the caital letters, D and L, are used to emhasize that the ulse duration corresonds to the maximum extent of the uniform distribution. The initial condition for L is chosen to match the excitation laser duration, as reviously described. The 1-dimensional mean field model (MF1D) utilizes eqn (3) only with a constant radius, while the couled eqn (3) and (4) are solved simultaneously to give a 2-dimensional result (MF2D). The advantage of the mean field model is that the equations of motion for the ulse can be raidly integrated for a variety of initial conditions and give a readily estimate of the ulse size as a function of time. To date, the mean field model has been used rimarily to model freely drifting electron bunches. While several additions to the theory have been made, e.g. to model the electron bunch inside an electrostatic acceleration field, 9,15 no attemts have been made to model the effect of a magnetic lens on the electron ulse shae. Since these fields are resent in all electron diffraction and imaging instruments, this extension to the mean field theory has to be made to validate its redictive ower. 2. Mean field theory including lens system. To imlement the acceleration field inside the electron gun, the mean field theory can be extended by treating the ositions at the front and the rear end of electron acket searately and exlicitly, as F(t) and B(t). 9,15 The maximum longitudinal extent is then given by L(t)=F(t) B(t). Aroximating the focusing lens as a radial deceleration element, the equation of motion becomes d dfðtþ g dt F ¼ a A þ a X ðt t birth;front Þ dt d dbðtþ g dt B ¼ a A a X ðt t birth;rear Þ ð6þ dt d dt g C drðtþ dt ¼ a R a M where g i is the relativistic Lorentz factor for front, rear, and center, resectively, a X and a R are the axial and the radial accelerations as in eqn (3) and (4), a A is the electrostatic ð3þ acceleration in the electron source, and a M is the deceleration term due to the focusing lens, exressed as a Gaussian function to simulate the finite thickness of a magnetic lens. It should mentioned that the axial forces are resent only after the births of the front and the rear and that the number of electron is now a function of time, which is zero before the birth of the front, N after the birth of the rear, and linearly increasing in between to mimic the generation of the hotoelectrons. 3. Monte Carlo simulation (MC). In contrast to the mean field model, which treats the electron ulse as a continuous charge distribution, a N-body Monte Carlo simulation treats the electron ulse as an ensemble of N randomly generated, discrete articles. 12 To this end, we have develoed our own electron bunch roagation code, in which each article in the bunch moves under the influence of three distinct forces: (1) the electrostatic force of acceleration, (2) the magnetostatic force of the focusing lens, and (3) the Coulomb force for each of the N(N 1)/2 air-wise interactions within the bunch. The magnetic lens can be simulated by either (a) a sum of current loos, (b) the finite-sized coil aroximation or (c) by imorting an externally simulated field. Further elements such as the onderomotive force, gravitation, the earth magnetic field, stigmators coils, and time-deendent streak lates can be straightforwardly imlemented, if desired. Using this hysically more realistic model, it is ossible to roagate the electron ulse over its entire lifetime, i.e. from the birth of the individual hotoelectrons at the hotocathode surface to their arrival at the detector. The nascent hotoelectrons are randomly generated at the hotocathode with a Gaussian temoral and uniform satial distributions to account for the fact that the extraction inhole in the anode acts as a satial filter roducing an initially well defined lateral rofile. The direction of the initial electron momentum vectors are given by a cos 2 y distribution and the initial momentum distribution is chosen to be uniform from zero to a high energy cutoff corresonding to hn W, where W is the effective material-secific work function in the resence of a DC electric field. 16 The treatment of the individual electron trajectories is entirely relativistic and should reroduce the true ulse trajectory, rovided that the initial conditions are chosen accurately and the Monte Carlo samling is fine-grained enough that the results converge. The drawback of this method is its high demand of comutational time, which increases in roortion to the number of air-wise interactions calculated at each time ste. To ease comutational demand, we treated the ulse as an ensemble of reresentative articles of aroriately scaled charge and mass in order to model ulses containing more than 1000 electrons in a reasonable amount of time. We found that this level of Monte Carlo samling was sufficient to achieve convergence to within 10%. III. Results and discussion A. Temoral resolution 1. Initial energy sread. In order to avoid ulse broadening due to Coulomb reulsion and achieve ultrafast temoral This journal is c the Owner Societies 2008 Phys.Chem.Chem.Phys., 2008, 10,

5 resolution, the ultrafast electron microscoe has been designed to oerate in the zero-current limit, meaning that the column contains one or a few electrons at a time. For an instrument of this tye, the ultimate resolution is mainly determined by the excitation laser ulse length and by the initial kinetic energy sread of the hotoelectrons. The ulse broadening in the absence qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of sace charge is calculated from Dt ¼ Dt 2 hn þ Dt2 KE with Dt KE given by eqn (1) or eqn (2). Using instrumental arameters, the results are shown in Fig. 3 for Dt FWHM vs. DE i.we invoke, for simlicity, a uniform hotoelectron distribution (see ref. 17) centered at E i and the width increase given by DE i =2E i. The exact solution (solid lines, eqn (1)) reresents the broadening of the ulses during their flight to the interaction region, while the aroximate solution (dashed lines, eqn (2)) reresents the broadening in the acceleration region. Since eqn (1) and (2) give very similar results, the broadening in the field-free drift region is relatively insignificant. As evident from eqn (2), it would be desirable to roduce hotoelectrons with a narrow energy distribution, but a large mean kinetic energy to reduce the magnitude of the factor DE i = ffiffiffiffi E i. Exerimentally, however, the distribution of hotoelectrons always extends to the limit of zero kinetic energy, because electrons below the Fermi level may be hoto-emitted as well. 17 To achieve the narrowest ossible energy distribution, the hotocathode needs to be oerated at a low temerature and the hoton energy needs to be matched to the work function of the metal. 18 This aroach is taken in our instruments of UED and UEC, resulting in an estimated kinetic energy sread of o0.3 ev; in ref. 18, the exerimental DE i under similar conditions (for a gold hotocathode) was measured to be 0.1 ev. We consider the electron ulses roduced in UED4 and UEC first. The acceleration ga in these guns is ket as small as ossible to maximize the extraction field. Under the influence of a field strength of 20 MV m 1 and 10 MV m 1 for UED4 and UEC, resectively, the electron ulses are barely elongated by the resence of an initial kinetic energy sread. In contrast, the broadening is more ronounced in UEM1 due to the smaller acceleration field strength of 4.8 MV m 1. The UEM1 column, originally designed to oerate in continuouswave mode, features an acceleration ga one order of magnitude larger than the ga found in the home-built guns of UEC and UED. Nonetheless, even at DE i = 0.3 ev, the ulse length is only 300 fs. Since UEM oerates in the absence of sace charge, the hotoelectric energy sread resents the main contribution to the electron ulse broadening. If necessary, the electron ulse duration could be reduced to the excitation laser ulse length by an aroriately designed extraction module. The Fig. 3 Temoral broadening of the electron ulse due excess energy of the electrons above the hotocathode work function for the UED4 (blue), UEC (green) and UEM1 (red) instruments. The ulse length at the secimen interaction region is comuted in the absence of sace charge using the instruments secific arameters. In the inset, the red arrow indicates the excitation from near the Fermi surface (E F ) to just above the vacuum, while the blue arrow is for excitation which carries an excess of hoton energy Phys.Chem.Chem.Phys., 2008, 10, This journal is c the Owner Societies 2008

6 consequence of the energy distribution, DE i, is in another asect of diffraction and imaging, namely the longitudinal coherence, which will be discussed below. 2. Sace charge effect. Before we evaluate by how much the ulse durations shown in Fig. 3 will be altered by the sace charge effect, we need to validate the different methods of simulating the electron acket. Fig. 4 shows the size (standard deviation) evolution of an electron bunch under the exclusive influence of Coulomb reulsions (DE i = 0), calculated using the mean field theory (i) in the longitudinal direction only (MF1D), (ii) in two dimensions with the imlementation of a focusing element (MF2D), (iii) in two dimensions with the imlementation of a focusing element and the acceleration field (MF2DA), and, ultimately, the Monte Carlo (MC) simulation. We note that s temoral in mm is the longitudinal extent of the sace charge limited electron acket and, knowing the seed (ost acceleration), can be exressed in the time domain using eqn (5). The initial conditions were chosen as follows: electrons er ulse, 30 kv acceleration voltage, Gaussian FWHM (110 fs) or the equivalent uniform rofiles in time, uniform rofile in the lateral direction (r = 100 mm). In each case, the current of the magnetic lens was otimized, such that the electron beam would have the smallest beam waist as it hits the detector. Not surrisingly, the MF1D aroximation with a fixed radius clearly overestimates the longitudinal sreading of the ulse, while the other methods give a good agreement. The radial acceleration used in the mean field models (eqn (4)) is only valid for infinitely thin disks and, therefore, overestimates the sreading in the lateral direction as the ulse elongates, such that even after otimizing the magnetic lens, the radius remains too large. In site of this large error, the longitudinal sread is only affected slightly, at least at this articular charge density. We can conclude that the mean field model may give a reasonable estimate of the ulse duration for a sace charge limited beam for a choice of arbitrary initial conditions. However, the radial exansion is not accurately reroduced by the current model and would require modification of eqn (4). The extent of the sace charge-induced broadening is mainly determined by the magnitude of the Coulomb reulsions, as well as the time scale of this interaction. By emloying higher acceleration voltages, the effective roagation time of the electron ulse can be reduced, since the electrons arrive at the interaction region in a shorter time and the sace charge induced broadening has less time to act. We used the mean field model to estimate the ulse length in our instruments as a function of roagation distance. The extraction voltages are 30, 60 and 120 kv for UEC, UED4, and UEM1, resectively. The effect of the acceleration voltage on the temoral duration (Gaussian FWHM) of the ulses as a function of the roagation distance is shown in Fig. 5. It is clear that UEM1 in single-electron ulsed mode is not limited by the sace charge effect, while the ulses in UED4 and UEC are broadened to several s once they arrive at the interaction region. Alternatively, the time of the sace charge induced broadening could also be reduced by lacing the interaction region very close to the electron source. 19 To isolate the role of the initial charge density, we calculated the temoral extent of the electron ulses under identical acceleration conditions. The initial conditions were ket identical to the conditions used for the comarisons of the models in Fig. 4, while the number of electrons was increased by an Fig. 4 Comarison of the length (a) and the radius (b) of the electron ulse redicted by the mean field theories and the N-body Monte Carlo simulation in the absence of an initial kinetic energy sread. Fig. 5 Comarison of temoral broadening due to the sace charge effect (DE i = 0) as a function of the roagation distance in UEC (green line), UED4 (blue line), and UEM1 (red line) using MF2DA. This journal is c the Owner Societies 2008 Phys.Chem.Chem.Phys., 2008, 10,

7 Fig. 6 Comarison of total temoral broadening due to the sace charge effect after 2 ns of roagation using MF2DA (blue line), MC 0.1eV (red line), and MC 0.3eV (green line). Available exerimental data are given for UEC (blue dots), and UED3 (red dots); see Fig. 2 (UEC and UED3, 30 kv). order of magnitude at a time. Fig. 6 shows the theoretical (MF2DA and MC) ulse duration after 2 ns of roagation time together with the exerimental measurement for validation. 20 Both models reroduce the ulse broadening reasonably well in the region where exerimental data is available. Using the MC simulation, two curves were calculated for different initial kinetic energy sreads corresonding to 0.1 and 0.3 ev. There is a slight difference between these two calculations in the low electron density regime, where the sace charge effect does not lay a dominant role. However, the curves quickly begin to overla each other as the electron density grows and for a ulse containing as few as 1000 electrons, the sace charge-induced broadening already masks any contribution of the initial kinetic energy sread after 2 ns of roagation. The mean field model can reroduce the results from the MC simulation quite accurately, but a more ronounced deviation occurs at higher charge densities, since the error in eqn (4) is exacerbated in this regime. Again, the temoral resolution caability of the ulsed electron robe can be successfully estimated with the mean field model; however, the satial resolution caability remains uncertain. B. Satial resolution 1. Coherence. The evaluation of the satial resolution of the ulsed electron robe requires a detailed discussion of the coherence of the electron acket. Coherence is the degree of a hase relationshi, which can give rise to interference. In light otics, an aerture is often emloyed to generate a seudo oint source. For a single illuminated object, the analogue of the double-slit exeriment, the coherence length is defined as the maximum length beyond which the interference fringe is attenuated. Below, we will consider the effect of many, satially searated objects (interaction region). If the aerture is small and the distance to the object is far, then the coherence length is defined as r c = l/(2a), where a is the half-angle subtended by the aerture. In such a case, the object is illuminated by sherical waves emanating from every oint of the aerture. It should be mentioned that the criterion r c = l/(2a) corresonds to only a 12% reduction of the erfectly coherent visibility. 21 This definition holds true, only as long as a is smaller than the hoton s intrinsic divergence, da, which can be estimated using the uncertainty rincile: da D r h l z 2Dx h ¼ l ð7þ 4Dx where r and z are the hoton momenta in the radial and longitudinal direction, resectively, and Dx is the aerture dimension. However, when a is bigger than da, then the object is illuminated only by an area within the angle da and the contribution from the rest of the source can be neglected. In such a case, the coherence length should be defined as r c = l/(2da). In contrast to hotons, free electrons are generated with an initial momentum sread, which determines the intrinsic divergence da for each electron, since the contribution originating from the uncertainty rincile term is negligible due to the small de Broglie wavelength. Using an acceleration voltage of 60 kv and assuming DE i E 2E i E 0.3 ev, the value of da ¼ D r f D i f sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 0 2E i ðde i Þ 2 2m 0 E f ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffi ðde i Þ 2 4E i E f becomes 1.6 mrad, which is an uer limit value. (The resence of a da ultimately gives rise to a non-zero transverse beam emittance. 22 ) The coherence length defined above only alies for a single scattering object. Blurring of the interference fringes needs to be taken into account in the case of scattering from objects with satial extent as in molecules or in the condensed hase. In this context, it is more aroriate to derive the coherence length in terms of interference fringe blurring. During the elastic scattering rocess, the magnitude of the momentum transfer is given by s ¼jk!! k 0j¼ 4 l sin y ð9þ 2 where k ~ 0 and k ~ are incident and scattered wave vectors of the electron, l is the electron s de Broglie wavelength, and y is the scattering angle. When the ensemble of electrons and the interaction volume have a finite size, each electron has an uncertainty in its osition, when it scatters. The error in the observable momentum transfer at a given radial osition s on the observation lane can be evaluated by ds ¼ s 2 dy 2 2 dl 2 @x ð8þ ð10þ The dy term reresents the uncertainty in the scattering angle, originating from the incident angular sread at each osition in the interaction volume (lateral coherence length), the dl term reresents the longitudinal energy sread (longitudinal coherence length) and the dx, dy and dz terms are reflective of the 3-dimensional size of the interaction region and the electron bunch. The db term is an uncertainty in measuring the momentum transfer due to the electron detection rocess, e.g. through blurring and binning the image (signal converter, 2900 Phys.Chem.Chem.Phys., 2008, 10, This journal is c the Owner Societies 2008

8 amlifier, and digitizer). In case of gas-hase electron diffraction, each distinct internuclear distance roduces sinusoidal interference fringes, 23 while in crystallograhy, reeated longrange order of scattering centers roduces a series of Bragg sots with a corresonding frequency in sacing. 24 The uncertainty in s will reduce the visibility of the interference fringes or the Bragg sots, such that they become unobservable and indistinguishable (see Aendix E). The instrumental coherence length can then be defined as R c ¼ 1 ds 1 2s s ð11þ We note that this definition reduces to r c = l/(2da), the lateral coherence length, only in the single scattering center limit and assuming the detection rocess is erfect and dl is negligible. Additionally, from discrete Fourier transformation theory, it is known that the maximum resolvable distance after collecting discrete data in the frequency domain, is given by r max ¼ 1 Ds ¼ 1 ð12þ s iþ1 s i Therefore, these values give an estimate of the maximum distance that can be decisively resolved in an electron diffraction exeriment. In the high resolution imaging mode of the transmission electron microscoe, the coherence of the electron ensemble is exloited to form contrast in the image. Concetually, the same limitations as stated above aly for the instrument s satial resolution. Since each electron interferes only with itself, the image is comosed of the indeendent suerosition of single electron interferences and image blurring occurs in the resence of an uncertainty in the electrons trajectory and their wavelength. However, in the more comlicated microscoe column additional resolution limiting factors have to be considered. In the wave-otical theory of imaging, 25 contrast in the image is formed as dictated by the hase contrast transfer function (CTF), which is a function not only of the roerties of the electron beam (incidence angle sread and wavelength sread) but also of instrumental arameters such as the sherical and chromatic lens aberrations, as well as aerture sizes and the articular defocus setting. Most commonly used is the Scherzer defocus, which, in combination with an aroriate aerture size, maximizes the contrast and allows for easily interretable images. The Scherzer resolution can be imroved uon by exit wave reconstruction techniques using images collected at different defocus settings, but the ultimate information limit of the instrument remains limited by the beam roerties and the quality of the electron otics. Since UEM oerates outside of the sace charge limit, the satial resolution achievable is the same as is obtained in continuous-wave mode and we will not comment on it further. 2. Otimal diffraction geometry. In conventional diffraction hysics, 26 it has been established that ds is minimized, when, for each electron, the incident wave vector, ~ k 0, is coincident with a line from the electron s osition in interaction volume to the center of detector. As is shown schematically in Fig. 7, the most extensive blurring occurs, if the beam arrives at the Fig. 7 Effect of the beam geometry on interference blurring for a finite sized beam using diverging (a), collimated (b), or converging (c) electron trajectories. interaction region on a diverging trajectory, i.e. the articles radial divergence angle, k = tan 1 ( r / z )40. On the other hand, if the electrons are erfectly collimated (k = 0), then the blurring on the detector is identical to the size of the electron beam waist. The otimal resolution is achieved, if the electrons are focused to a oint on the center of the detector and iminge on the interaction region on an ideally-focused converging (k o 0) trajectory. It follows that for a given camera length, the ideal radial convergence angle, k, in the interaction region is given by an aroximately linear function of the off-axis distance. Using an interaction volume with dimensions dx = dy = dz = 300 mm, an intrinsic electron divergence dy = 0.5 mrad, a wavelength sread dl corresonding to a kinetic energy sread of 0.3 ev, and a detector blurring db = 100 mm, a coherence length of B12 A is achieved in the converging beam configuration, while the collimated beam and the diverging beam only give B4 and B3 A, resectively. The dl term in eqn (10) is much smaller comared to the remaining terms, such that the blurring of the interference fringes is not affected by longitudinal coherence. 3. Focusing behavior. In the absence of Coulomb reulsion, dy, which originated from the intrinsic divergence, da, will lead to a finite sot size on the detector. Therefore, the ability to focus the beam to a small oint on the detector can be a direct measure of the instrumental coherence length, because ds is dominated by dy after all other terms have been minimized. For high energy electrons, the de Broglie wavelength is much smaller than the source dimension and we can rather treat each electron in the bunch as a classical article and its trajectory as a ray, which is influenced by external forces, if any. However, due to the resence of da, the focusing behavior of the electron beam becomes less than ideal, as shown schematically in Fig. 8. Assuming that the electrons are generated in a source of radius R 0, having an overall beam divergence O 0, if any, and further assuming that each emitting oint in the source inherently diverges with an angle da after acceleration, then these two comonents are focused at different ositions. The focal distances, B and b, roduced by a lens of a focal length, f, for the O 0 and da comonents, resectively, This journal is c the Owner Societies 2008 Phys.Chem.Chem.Phys., 2008, 10,

9 Fig. 8 Radial focusing behavior of a finite-sized beam in the absence of sace charge. are given by the lens equation: 1 f ¼ 1 A þ 1 B for O 0 1 f ¼ 1 a þ 1 for da b ð13þ The da comonents are erfectly focused at the imaging lane b, where a magnified image of the source can be formed, while the smallest overall beam waist is obtained near the focal lane B. If the radius due to each comonent (da and O 0 ) can be determined indeendently from the two different focal distances, then the resulting beam size along the beam ath may be comuted by convoluting the radii of these two comonents, i.e. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RðxÞ ¼ R O0 ðxþ 2 þ r da ðxþ 2 ð14þ where A B x R O0 ðxþ ¼ R 0 A a B ¼ R A 0 A a 1 þ x A x f ð15þ and r da ðxþ ¼ a tan da b x 2 b ¼ a tan da 1 þ x 2 a x f ð16þ and O 0 A ¼ a þ R 0 cot ð17þ 2 The otimal focal length of the lens can then be determined by minimizing R with resect to f. Fig. 9a shows a result of eqn (14), (15), and (16) using tyical UED arameters. It can be readily seen that, even though the O 0 comonent is focused, the minimized sot size is finite and effectively determined by the diverging da comonent. Sufficiently small radii can be obtained either by (i) making R 0 very small (oint source), or (ii) by making O 0 large. Both aroaches essentially bring the virtual source osition, A, closer to the actual source lane, a. Fig. 9 Evolution of the beam waist calculated using the two-comonent convolution model of eqn (14) (a), and deendence of the focal sot size on the initial source dimensions (b). Fig. 9b shows how the final focus size deends on the initial source dimension. Additionally, a smaller focal size can be obtained by reducing the intrinsic divergence da by using higher acceleration otentials. In conventional electron beam sources, a small robe size can be readily obtained by using a nanometer scale field emission ti in combination with a high acceleration voltage. 25,27 In the high current limit of ulsed electron guns, however, where bunches contain thousands or ossibly millions of electrons, a finite-sized source becomes a necessity due to extraction quantum efficiency of the cathode material. Therefore, the electron beam has to be given a macroscoic divergence, O 0, if a small focus size is desired. Exerimentally, this can be accomlished by utilizing the negative lensing effect of Coulomb reulsions (which deends on the initial charge density) and/or by inserting a diffusive lens immediately after the source to controllably induce this divergence. For ulsed electron guns in the sace charge limit, any effort to minimize the hotoelectric momentum sread is inconsequential, since generation of a sufficiently cold beam does not totally eliminate the da comonent. The random and discrete nature of the electron s osition within the ulse and the corresonding irregular Coulomb reulsions can also roduce an intrinsic divergence for each electron. An estimation of its magnitude can be made as follows: When the 2902 Phys.Chem.Chem.Phys., 2008, 10, This journal is c the Owner Societies 2008

10 robability of electron to be at osition ~r is given by P(~r), the mean Coulomb otential energy becomes Z 1 e Vðr! iþ¼ 2 4e! 0 r! NPðr! Þdr! ð18þ r i and its deviation is given by Z 1 e 2 dvðr! iþ¼ 4e! 0 r! d NPðr! Þ dr! r i ð19þ which can be aroximated to give Z 1 e dvðr! iþ 2 4e! 0 r! NdPðr! Þdr! r i vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ne ub 2 C 4e 0 hr! iþdr!! A 1 Ne 2 C r i 4e hr! i r! A i 1 Ne 2 dr 4e 0 r 2 i ð20þ Thus, dv(~r i ) becomes B0.7 ev for r = 1 mm and N =10 6. MC simulations confirm that, when 10 6 electrons are generated within a shere of 1 mm radius, an angular deviation of da = 0.5 mrad develos after a few nanoseconds, which is equivalent to an initial hotoelectric kinetic energy sread of B0.2 ev. Fig. 10 Monte Carlo simulations of the radius (a) and the ulse length (b) of a bunch containing 10 6 electrons using UED4 instrumental arameters. 4. Satial resolution for sace charge limited ulses. Unlike in continuous beam diffraction, a well-focused converging beam of small size is hard to achieve when using sace charge limited electron ulses as a robe. In the high charge density regime, the smallest robe size achievable is limited by the magnitude of the Coulomb reulsion among electrons. Consequently, it becomes imossible to focus the beam to a small oint on the detector. However, the actual sot size of the unscattered beam is not imortant, as long as the Coulomb reulsions do not alter the converging electron trajectories before the ulse has assed the interaction region. Once the ulse has assed the interaction region, the scattered electrons searate from the main beam and carry the information about the secimen (encoded in the scattering angle) to the detector. Coulomb reulsions do not affect these trajectories, since the fraction of scattered electrons is small and the scattering angles are large comared to the divergence angle of the unscattered beam. This is in contrast to the imaging mode in the electron microscoe, where the signal carrying electrons are focused again after the secimen. At the high eak currents in singleulse oeration, these focusing rocesses can lead to trajectory dislacements through random scattering of the imaging electrons and, subsequently, to a stochastic reduction of the image resolution. 28 To investigate a ulsed beam s satial resolution caability, it is necessary to obtain more detailed insights into the electron bunch roerties than can be obtained from the mean field model. The N-body Monte Carlo simulation delivers the 3- dimensional osition and the 3-dimensional momentum vectors for each of the articles in the bunch and is an ideal tool to study the coherence of the ulsed electron robe. We simulated electron ulses containing 10 6 electrons using an arbitrary gun design for a tyical UED exeriment. Fig. 10a and b show the evolution of the ulse radius and duration, resectively, with four different initial conditions, which determine the extent of the initial electron acket. It is evident that a satially and temorally confined ulse goes through an initial Coulomb exlosion and exands vigorously in both the lateral and the longitudinal direction. The lateral exansion can be comensated for by the magnetic lens, but the longitudinal exansion is unaffected by the focusing field resulting in a larger temoral extent of the ulse at later times. After the initial Coulomb exlosion, the ulse exansion is aroximately linear, indicating that the sace charge effect has ceased to alter the ulse exansion in the drift region. The same cannot be said for the other ulses, where the stress of high initial otential energy had been alleviated by initially stretching or exanding the excitation laser ulse in either the lateral or the longitudinal direction or both. For these ulses, the evolution of the radius and the length remains uward curved (with the excetion of the lateral exansion in the region of magnetic lens focusing) for the entire drift region. All but one of the electron ulses collide with the samle as a diverging bunch, which, as was shown above, is detrimental to the satial resolution attainable in the instrument. This journal is c the Owner Societies 2008 Phys.Chem.Chem.Phys., 2008, 10,

11 Fig. 11 shows the lateral hase sace rojection of the four ulses at longitudinal ositions of (a) z = 100 mm, (b) z = 300 mm, (c) z = 500 mm, and (d) z = 645 mm (interaction region): the articles radial divergence angle, k = tan 1 ( r / z ), defined as the angle directing the electron toward (k o 0) or away from (k 4 0) the otical axis as a function of its radial distance from the otical axis. The initially ositive, i.e. radially diverging, chir is reversed in direction by the magnetic lens. After assing the magnetic lens, the electrons once again come in close roximity further downstream in the region of the smallest beam waist. In this region of high charge density, Coulomb reulsions gradually reverse the sign of the linear correlation between radial ositions and the convergence angles until a new diverging chir has develoed. The ideal convergence angle, shown as the black line in Fig. 11d, is only reroduced by the initially well-confined ulse, since this ulse was able to exand initially and escae the shere of influence of detrimental sace charge effects. For this ulse, dy = 1 mrad and the resulting coherence length is 7 A, using the reviously stated uncertainties for the other terms in eqn (10) and (11). This articular ulse, although caable of roducing high resolution information, will result in a reduced signal (for gas hase scattering), since the robability of the scattering events is directly roortional to the integrated areal density of the scattering centers, as well as the number of electrons assing through the interaction region. The signal intensity is increased, if the cross sectional area of both the electron beam and the interaction region is reduced in size (assuming the samle delivery rate is constant). The results in Fig. 10 and 11 suggest that in order to obtain a small robe size and the corresonding convergence angles at the interaction region, the electron ulse has to avoid excessive Coulomb reulsion until the electrons have scattered from the secimen in the interaction region. Consequently, for a beam of free electrons, the high charge density has to be relieved by stretching the ulse in the longitudinal direction, such that the trajectories of the electrons are not altered by Coulomb reulsion before the interaction region and the ulse remains ideally focused. In other words, if the molecular density is low (e.g. in gas hase diffraction) or the signal averaging time is limited such that the diffraction image has to be acquired in a single shot, then good satial resolution and good signal intensity can only come at the exense of temoral resolution. On the other hand, if the exeriment can afford an intermediate number of electrons ( ), then the tradeoff between temoral and satial resolution becomes less demanding. To quantify these statements, we otimized instrumental arameters to obtain a smaller beam waist at the interaction region. To roduce the required convergence angle, the Fig. 11 Radial divergence angles of individual electrons. The results are for the ulses shown in Fig. 10 at axial distances of z = 100 mm (a), z = 300 mm (b), z = 500 mm (c), and z = 645 mm (d). All ulses develo a diverging chir (k = tan 1 ( r / z )40) due to sace charge. This linear correlation is reversed in sign (k o 0) by the magnetic lens, a condition necessary for converging beam diffraction. However, the sace charge effect alters the converging electron trajectories for three of the four ulses in the figure by the time they arrive at the interaction region. Only the initially-confined ulse (shown in blue), which undergoes a Coulomb exlosion at early times, is able to escae the shere of influence of the sace charge effect and reroduce the ideal convergence angle (black line) in the interaction region Phys.Chem.Chem.Phys., 2008, 10, This journal is c the Owner Societies 2008

12 Fig. 12 Radial divergence angles of individual electrons using original (blue) and otimized (red) geometries. The uncertainty in scattering angle directly results from the deviation of incident angle from the ideal convergence angle. The red and blue shaded areas cover articles falling within an angular sread of s. The mean field theory assumes a erfect correlation between electron momenta and osition, such that dy = 0. Itis evident that, in reality, electron trajectories are not erfectly correlated (dy 4 0). The otimized instrumental geometry imroves the coherence of the beam by reducing dy and, additionally, gives a smaller robe size (see text). electron beam must have exanded laterally by the time it is being focused by the magnetic lens. If, however, the ulse is initially stretched in time to say B100 s, then, for this rolate ulse, the intrinsic rate of the lateral exansion is too small to roduce a beam big enough. Exerimentally, a larger beam size at the magnetic lens osition can be realized in three ways: (i) by using a diverging lens in the source region, (ii) by increasing the acceleration ga, or (iii) by lacing the magnetic lens farther away from the hotocathode. In the last two instances, which deend on the initial charge density, the ulse is given more time to exand before it is being focused toward the interaction region, i.e. the sace charge effect is used as a diverging lens. Interestingly, a longer acceleration region must not necessarily have a detrimental effect on the temoral exansion of the electron ulse, as was reviously ostulated. 29 Since the early events in the lifetime of the electron ulse determine its future behavior, this measure can, under certain conditions, bring about a reduction in the longitudinal momentum sread (see Aendix F). Using the second aroach to exand the beam, a ulse containing 10 6 electrons giving imroved satial resolution comared to the revious ulses is shown in Fig. 12. The exerimental conditions in this case are the following: A uniform lateral rofile with a small initial radius (r = 100 mm), a broad Gaussian rofile in time (Dt = 100 s), a large acceleration ga (d = 25 mm), a magnetic lens located at z = 180 mm, and a shorter drift sace to the interaction region at z = 300 mm. For this ulse, dy = 0.5 mrad and the resulting coherence length is 12 A, using the reviously stated uncertainties for the other terms in eqn (10) and (11). Alternatively, a small robe size could also be realized, if the ulse started out with a big initial diameter. However, we found that due to sherical aberration of the magnetic lens, this aroach was less successful. It should be noted that the otimized ulse shown in Fig. 12, although being better than the ulse under the original design, might not be the best ulse given the multitude of exerimental arameters. The search for the otimal configuration would involve minimizing the ulse waist and the convergence angles to accetable values, by simultaneously changing several exerimental arameters, including the electron ulses initial length and width, the length of the acceleration ga, the osition of the magnetic lens, and the drift length to the interaction region. Summary and conclusion The electron roagation dynamics determine the satiotemoral resolution of diffraction of imaging. In this contribution, This journal is c the Owner Societies 2008 Phys.Chem.Chem.Phys., 2008, 10,