Mechanisms of decoherence in electron microscopy

Ultramicroscopy 111 (2011) 761–767

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Mechanisms of decoherence in electron microscopy A. Howie Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, UK

a r t i c l e in fo

abstract

Available online 30 July 2010

The understanding and where possible the minimisation of decoherence mechanisms in electron microscopy were first studied in plasmon loss, diffraction contrast images but are of even more acute relevance in high resolution TEM phase contrast imaging and electron holography. With the development of phase retrieval techniques they merit further attention particularly when their effect cannot be eliminated by currently available energy filters. The roles of electronic excitation, thermal diffuse scattering, transition radiation and bremsstrahlung are examined here not only in the specimen but also in the electron optical column. Terahertz-range aloof beam electronic excitation appears to account satisfactorily for recent observations of decoherence in electron holography. An apparent low frequency divergence can emerge for the calculated classical bremsstrahlung event probability but can be ignored for photon wavelengths exceeding the required coherence distance or path lengths in the equipment. Most bremsstrahlung event probabilities are negligibly important except possibly in largeangle bending magnets or mandolin systems. A more reliable procedure for subtracting thermal diffuse scattering from diffraction pattern intensities is proposed. & 2010 Elsevier B.V. All rights reserved.

Keywords: Electron decoherence Bremsstrahlung Thermal scattering

1. Introduction Preservation of coherence in strongly interacting situations is a widely appreciated personal talent that is frequently manifested by our esteemed colleague John Spence. In the world of electron microscopy its crucial role is equally marvellous and well recognised in making possible the observation of many interference effects such as diffraction contrast imaging and in even more demanding situations, phase contrast imaging and electron holography. For interactions associated with measurable energy losses, energy selection or energy filtering techniques are now routinely available to identify their precise role and if necessary restrict their contribution to one of image attenuations rather than decoherence. At lower energy transfers, imaging apertures can take on these selection or filtering functions in the case of large-angle scattering events such as thermal diffuse scattering. The effect of various electron beam-specimen interactions is considered for internal beams in Section 2 and for external aloof beams in Section 3. Particular attention is paid to small-angle scattering events associated with energy losses too small to measure. In the context of achieving high spatial resolution, the significance of such decoherence events through the influence of time-dependent stray electromagnetic fields and the need for adequate screening has long been recognised in the electron optics. In Section 4 the additional possibility is addressed that

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significant decoherence may be caused by the emission of bremsstrahlung radiation generated in the electron optical column following the increasing use of elaborate multipolar aberration correction systems, spectrometers and monochromators. Phase retrieval techniques applied to diffraction data would aim to dispense with at least some of this electron optical sophistication but may require their own special consideration for the treatment of thermal diffuse scattering as is investigated in Section 5.

2. Material interactions with penetrating beams 2.1. Electronic excitation With a typical mean free path by fast electrons of about 100 nm, valence excitation, particularly of plasmons or collective modes, is usually significant in transmission electron microscopy. The realisation [1] that many of the diffraction contrast interference effects were being observed with electrons that might have suffered at least one such inelastic event was initially puzzling in view of the associated change in electron wavelength and frequency as well as the unknowable phase shift in inelastic transitions. The explanation [2,3] depends firstly on the fact that each energy component of the electron wave can interfere only with itself if a static pattern is to be observed and secondly that the different spatial components (or electron paths) at each energy must in further loss processes generate the same specimen

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excitation (thus suffering identical if unknown phase shifts). Here and in some following sections, it is convenient to consider scattering transitions for the simplest wave functions describing fast electrons travelling close to the z-axis and conveying coherent information about for example a crystal lattice image:

C1 ¼ exp½ikð1Þ :rcosðgx=2Þ C2 ¼ exp½ikð2Þ :rsinðgx=2Þ

ð1Þ

A linear combination of these two Bloch functions can be used to describe an electron of energy E propagating at the Bragg condition in a crystal. In this case k(1) and k(2) both lie in the z-direction but, because of the crystal potential, have slightly different magnitudes so that interference gives rise to thickness fringes in a wedge crystal. The interference term allowing crystal lattice imaging is present in a different form in each case. The dispersion surface diagram of Fig. 1 shows more generally the bands for energy E with the behaviour of k(1) and k(2) (represented, respectively, by points A and B) near the Bragg condition. After inelastic scattering, an electron has to be described by a dispersion surface for its new energy E0 ¼E DE with wave points such as C and D depending on the momentum transfer :q involved. At the energies employed in transmission electron microscopy, these two sets of dispersion surfaces are to an excellent approximation simply separated by a small displacement qz ¼ DE/:v¼ o/v in the kz-direction so that the same transition takes the wave points A and B to C and D. Any interference effect between A and B is thus replicated faithfully between C and D. A crucial requirement here is that only intraband scattering events A-C and B-D occur and that interband transitions such as A-D or B-C are forbidden. With the exception of light emission events discussed below, such interband transitions are umklapp processes corresponding to momentum transfer :(q+g) where g is a reciprocal lattice vector. They will not arise at small q vectors for a highly delocalised, longitudinal Coulomb excitation like a plasmon described by a simple plane wave q without any modulation q+g due to the crystal lattice. In practice the degree of similarity between diffraction contrast images for different plasmon losses is very high but begins to fail even within the crystal due to effects of finite scattering angle with associated changes in Bloch wave transitions and propagation characteristics [4]. The finite scattering angle effect appears when the defocus dependence of the images is studied as shown in Fig. 2 adapted from [5] (see also [6]). With defocus distance Dz, the coherence pffiffiffiffiffiffiffiffiffiffiffiffi of the electron wave is tested over a lateral distance x  ðlDzÞ. Although the zero loss and plasmon loss thickness fringe images are quite similar in focus,

A E B

C E' D

Fig. 1. Dispersion surfaces for an electron of energy E (heavy lines) and lower energy E0 (light lines). Possible inelastic transitions are represented by A-C, B-C, A-D or B-D.

at Dz¼200 mm, xE25 nm and approaches the lateral coherence distance of the excited plasmon wave packet (also reflected in the angular dependence of inelastic scattering) and so the plasmon image contrast becomes much weaker. The increased loss of contrast of plasmon images compared with zero loss images was also noted for the defocus conditions used in Lorentz microscopy of magnetic domain structures [7]. Near-focus images such as those in Fig. 2(a) or typical high resolution electron microscope (HREM) images are in fact in-line holograms. At large defocus values such as in Fig. 2(b), they are clearly somewhat more like off-axis holograms. General discussions on the possibility of electron holography with inelastically scattered electrons [8] show that a necessary requirement is that the electron wave on each path should excite the same event. Clearly this will not be possible in the commonly used situation with the reference beam well outside the specimen and the second beam path inside. However with both beams inside the specimen but close enough to excite the same plasmon, off-axis inelastic electron holography is indeed possible and has recently been impressively applied to investigate plasmon coherence with results that seem consistent with Fig. 2 in terms of their dependence on shear (the lateral separation of the two paths) [9]. By looking at different materials, the dependence on plasmon energy was also measured. As discussed in Section 3 below, these inelastic holography experiments were further extended to the aloof beam situation, where both beams are outside the sample but close enough for Coulomb interaction to generate a plasmon [9]. 2.2. Phonon excitation Phonon excitation or thermal diffuse scattering (TDS) can be even more probable than plasmon excitation in electron microscopy but involves an umklapp interaction corresponding to interband transitions and predominantly large-angle scattering well outside the typical imaging aperture. This process has long been well understood as the source of anomalous absorption in transmission electron microscopy and in recent decades as a primary contrast mechanism in high-angle annular dark field (HAADF) incoherent imaging in the scanning transmission electron microscope (STEM). The small TDS fraction accepted by the microscope aperture can in principle contribute to HREM images but its precise role has been difficult to study experimentally. Comparison of images with and without the TDS contribution has however been achieved by off-axis electron holography, indicating that near focus the former may actually exhibit contrast subtracting from the elastic scattering contrast in the traditional under focus positive Cs condition and enhancing it in the overfocus negative Cs condition [10]. Nevertheless, with increase in defocus, a TDS decoherence effect could be expected to develop much faster than for the small-angle scattering case of plasmon excitation. This so far imperfectly resolved issue has relevance in addressing remaining discrepancies between observed image contrast and theoretical simulations (the Stobbs factor). Recent work claims successfully to have eliminated the Stobbs factor for both HAADF and bright field STEM images, leaving any residual uncertainty for conventional HREM images [11]. In the absence of excluding apertures, TDS can clearly play a more significant decoherence role in the diffraction pattern and this is considered in Section 5. 2.3. Radiation processes in materials The energy–momentum relations for electrons and photons, which preclude coupling in free space, can be modified in a solid to allow, respectively, bremsstrahlung and Cherenkov emission.

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Fig. 2. Intensity profiles of thickness fringes taken with zero loss electrons and plasmon loss electrons. The traces agree quite closely under focus conditions (a) but at 200 mm defocus (b) the contrast in the plasmon loss profile has largely disappeared. Images adapted from [5].

When elastic scattering solutions are available e.g. the Bloch wave for electron propagation in a crystal, the former process can conveniently be described in terms of energy loss by spontaneous emission driven by a semi-classical radiation potential Vsc(r,t) of the form [12].  1=2 e2 _ Vsc ðr,tÞ ¼ i exp½iðqUrotÞAUr ð2Þ 4pmce0 Here q and o are the wave vector and frequency of the emitted photon, respectively. The vector potential A takes the direction of the electric polarisation vector and for a light wave is normal to q in contrast with the interaction with longitudinal excitations described above. For instance the interband transition B-C in Fig. 1 allows the fast electron to lose energy :o with a photonmatching momentum change :o/c less than the minimum value of :o/v, which would apply in free space or for intraband transitions A-C or B-D [3,13,14]. For such interband processes involving reciprocal lattice vectors g in the zero-order Laue zone the emission (called channelling radiation) is from a dipole in the x,y plane and gives rise to distinct spectral peaks when the above energy momentum transfer conditions are matched. In the typical transmission microscopy situation, the channelling motion might have an amplitude A E0.1 nm, a wavelength of LE20 nm and a corresponding dipole frequency o ¼2pv/LE4  1016 Hz. The event probability (i.e. number of photons radiated per channelling cycle) can then be roughly estimated from the Larmor formula (see Appendix A) as Pchan ¼(e2/3e0:c)(Ao/c)2 E5  10  7. Each of these highly improbable events would be associated with a reduction of interference contrast since the gradient operator in Eq. (2) with its main component in the x-direction induces a change from sin(gx/2) to cos(gx/2) in the electron wave function. Coherent bremsstrahlung emission in the X-ray region can occur with still lower probability when reciprocal lattice vectors for other Laue zones are involved [13,14]. These crystal bremsstrahlung processes become important for electrons of much higher energy but have extremely small probability in electron microscopy. Furthermore the loss of electrons if necessary can easily be removed by energy filtering so they need not present a practical source of decoherence.

In the band gap region of a semiconductor or insulator with refractive index n(o), the velocity of light c/n(o) can be below the velocity of a fast electron making possible the emission of Cherenkov radiation. The probability PCh of Cherenkov radiation per unit path length by a fast electron of velocity v is given by the expression [15]  Z O e2 c2 PCh ¼ 1 ð3Þ do 4pe0 _c2 0 n2 ðoÞv2 Here the integral extends over the frequency region where the integrand is positive. For 200 keV electrons the mean free path for emission of a Cherenkov photon in typical cases can be about 5000 nm. In slab specimens, many of these photons will be confined in guided modes by total internal reflection although their contribution can still of course be detected in the loss spectrum except at the lowest energies. Yet another emission process is transition radiation associated with the sudden change in the electromagnetic field of a moving charge when it (or its image charge) crosses the boundary between two dielectric media. For normal incidence at the entrance surface to a perfect conductor, the photons can be emitted back into vacuum and, as outlined in the appendix, the emission probability P is crudely given by the expression Z O e2 v2 do Ptr ¼ ð4Þ 2 3 3p e0 _c 0 o For this simple model, the integral extends up to the plasmon frequency O where the metal becomes transparent. A more sophisticated analysis for a single interface can be found in [15]. In the typical thin film sample used in electron microscopy, the sources of transition radiation are the long wave modes excited at the entrance and exit surfaces, which can radiate for surface wave vectors q o o/c. In the case of a thin slab, the formation of symmetric and antisymmetric mode combinations as a result of coupling between the two surfaces takes care of the problem of multiple reflection in the emission of transition radiation. The resulting expressions for emission probability are inevitably quite complex [16] but still manifest the same low frequency divergence revealed in Eq. (4). This infrared catastrophe arises

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in a number of applications of classical electromagnetic theory and was initially resolved by invoking multiple photon emission processes [17] and later through higher-order corrections in quantum electrodynamics [18]. For the moment we note that to destroy electron interference by making a ‘‘which path’’ measurement using the emitted photons it would seem necessary to operate at wavelengths less than the maximum separation between the interference paths. This argument may suggest a plausible long wavelength cut-off but we return to the issue in Section 4 below when we encounter the infrared catastrophe once more. For a cut-off wavelength of 1 mm, Eq. (4) would yield a typical value Ptr ¼ 0.01.

3. Aloof beam excitations Using specially developed electron holography equipment operating at 1.65 keV and indicated schematically in Fig. 3, Sonnentag and Hasselbach [19] have made systematic measurements of decoherence effects when the two interference paths are passed at distance x0 from a semiconducting Si plate of resistivity 1.5 O cm. They attempted to fit the observed dependence of fringe visibility V on both x0 and Dy (the separation of the two paths) using the expression V¼exp( tflight/td) involving the time of flight over the plate tflight. The decoherence time td was related to the energy dissipation rate though the theoretical result for this had to be increased by a factor of about 100 to match the observations. In an alternative approach we may relate the decoherence effect not to energy loss but directly to event probability, for which expressions are by now well developed in the context of STEM probes in aloof beam mode [20]. For the simple case of a non-relativistic electron travelling in vacuum at velocity v parallel to and at distance x0 from a planar interface, the differential event probability for a path length L is given by the expression   d2 Pðx0 , o,qy Þ e2 L 1 exp½2qx0  ð5Þ ¼ Im 2 2 do dqy 1 þ eðoÞ q 2p e0 _v where q with components (0,qy,o/v) is the wave vector of the surface excitation. The effects of surface plasmons and similar excitations described by this equation can be removed by energy filtering but, when the electron travels for a considerable path length L near the surface of a conductor, events involving much lower energies could still be significant even at large impact parameters x0. In the very lowest frequency region we can replace the dielectric response function Im{  1/(1+ e(o)} by e0o/s, where s is DC electrical conductivity. Terahertz frequency characterisation of low carrier density semiconductors indicates that this

linear behaviour extends up to a frequency om of about 0.6  1012 Hz, above which the dielectric response function in Eq. (5) falls off quite rapidly [21]. For impact parameters x0 in the 10 mm range, the exponential factor in Eq. (5) also greatly reduces contributions from frequencies o 41012 Hz. Integrating Eq. (5) up to a cut-off frequency of om, we find " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# dPðx0 ,qy Þ o2m e2 L ð6Þ ¼ expð2x0 qy Þexp 2x0 þq2y dqy 4p2 _sx0 v2 To compute the decoherence effect we should now integrate Eq. (6) over the range of wave vectors 9qy94 a Dy  1 sufficient to distinguish between the two interference paths. Assuming that a Dy  1 4 om/v, we can make a binomial expansion of the second exponential term to obtain the result for the decoherence event probability P:    2 2 Z 1   expð2x0 qy Þ x0 e Lom 2ax0 P ¼ dqy ¼ 0:62E1 2 2 qy a Dy Dy 4p _sv a=Dy Z 1 expðxÞ dx ð7Þ E1 ðZÞ ¼ x Z Although P can exceed unity, the holographic fringe visibility V is given by V¼exp(  P). A noticeable feature of this result is that P depends not on x0 and Dy separately but simply on the ratio x0/Dy, which indeed seems to be roughly followed by the experimental data [19]. These data are plotted in Fig. 4 together with the function E1(4x0/Dy), indicating that with the very plausible value a ¼2, our relatively simple model can fit the observations to within a factor 2. For realistic path lengths near more highly conducting metallic bodies (s E107 O  1 m  1) such as the beam apertures encountered in an electron microscope column, it seems very unlikely on this simple ohmic loss model that these low frequency aloof beam excitations could contribute significantly to decoherence. Garcia de Abajo [22] has however recently computed the dephasing probability due to dipole radiation when a relativistic electron passes a perfectly conducting sphere. He calculates that a 200 keV electron passing 1 mm away from the surface of a 10 mm radius sphere experiences a dephasing probability as high as 0.3%. For passage through a metallic aperture Garcia de Abajo found a low frequency logarithmic divergence due to radiative processes rather similar to that already noted for transition radiation in Section 2. He treats the divergence by invoking the ‘‘which way’’ argument already mentioned that the lateral

4

3

D

Δy P

X 2

S Z

1

X0

Y L Fig. 3. Schematic diagram for electron holography decoherence studies with an aloof beam at distance x0 from a semiconducting plate. The two paths diverge from the point S and, after reaching a maximum separation Dy are then brought together again in a distance D to produce interference fringes. These trajectories can arise from propagation in a simple harmonic oscillator potential.

0

1

2

3

4

Fig. 4. Comparison between points for P ¼  ln(V), where V is the measured fringe visibility in aloof beam electron holography [19] and the theory of Eq. (7) represented by the function E1(Z), where Z ¼ 4x0/Dy.

A. Howie / Ultramicroscopy 111 (2011) 761–767

separation of interfering paths sets a lower bound for the emission frequencies that might cause decoherence. There is no divergence in the case of the conducting sphere, possibly because of the limited number of low frequency modes, but the ‘‘which way’’ argument might reduce the dephasing effect in this case also. The relevance of the lateral separation Dy of interfering paths is again clear from comparisons of zero loss and plasmon loss electron holograms obtained in aloof beam mode at various impact parameters x0 [9]. The fringe intensity and coherence of the zero loss hologram vary only slightly with impact parameter as would be expected from Eq. (5) for such a relatively short path length L near a high conductivity metal. For the hologram taken at the plasmon loss :op, the intensity falls off rapidly with roughly the modified Bessel function K0(2op/v) behaviour predicted by integrating Eq. (5) over qy. However coherence (fringe contrast) rises from a low value close to the surface, reaching a maximum near x0 ¼20 nm and falling gradually thereafter. As x0 is increased from 0, the exponential factor in Eq. (5) progressively reduces the contribution from the larger qy components, which would otherwise affect the two paths differently and reduce coherence. The very low intensity levels at still larger x0 values must make shot noise more important, leading to the observed fall off in apparent fringe contrast there. For a proper treatment of decoherence effects in these holography experiments the fast electron should clearly be described by appropriate wave functions before and after scattering rather than by a simple classical trajectory.

4. Decoherence due to bremsstrahlung induced in electron optics Bremsstrahlung emission has been most thoroughly investigated in the context of synchrotron emission by relativistic electrons, where the basic classical electromagnetic theory originally given by Schott [23] works extremely well. For such circular periodic motion, the emission takes place at multiples of the Larmor frequency oL ¼eH/m. In electron microscopy, the nearest approach to this situation occurs in components like the alpha or mandolin filter, where the beam makes approximately one complete loop in the magnetic field. Dividing by :oL the standard relativistic expression for the energy loss per turn [15], we get a rough estimate for the number of associated emission events of e2g4b2/(3e0c:) indicating about 0.05 for a 200 keV electron. Since electron velocity is unchanged after exactly one turn we do not expect any low frequency divergence of the type shown by Eq. (A4). The multiple photon processes which become dominant for highly relativistic electrons are noticeable even at 200 keV however and on the basis of the Schott theory [23] the above estimate for event probability should be reduced to about 0.03. The associated energy losses would of course be much too small to measure. Energy losses due to bremsstrahlung processes elsewhere in the electron optical column have not been reported and would not be expected to be significant. However, in cases where the electron experiences a change of velocity between the beginning and the end of the acceleration event, classical electromagnetic theory does exhibit in many cases a 1/o dependence for the radiation probability, leading to the infrared catastrophe already noted in Section 2 for transition radiation. A very simple example arises (see Appendix A) in a linear slowing down process such as would occur for instance in the dielectric excitation considered in Section 3 or more dramatically in an electron mirror. When the initial velocity v1 changes to a final velocity v2 in a distance short compared with the emission wavelength 2pc/o, the emission probability (or number of photons N emitted) per frequency range

765

is expressed by dPbrem dN 1 dI e2 ðv2 v1 Þ2 ¼ ¼ ¼ do _o do do 6p2 _e0 c3 o

ð8Þ

We can extend this result to cases where acceleration is not parallel to velocity by moving to a reference frame moving at constant velocity u, since neither N nor do/o will change. It is then clear that the logarithmic divergence will also be present in other cases such as a bending magnet when the electron beam suffers some deflection but no change in the magnitude of the velocity. With slight modifications (see Appendix A), Eq. (4) for transition radiation also follows from Eq. (8). Appendix A also provides support for the conjecture that the simple arguments used above to estimate emission event probability in one complete cycle of dipole or cyclotron emission may not require any large infrared correction since the initial and final velocities v1 and v2 are identical. A more detailed picture both of the decoherence contribution of bremsstrahlung emission in such a slowing down region and of an appropriate choice of cut-off to control the low frequency divergence can be obtained from a simple wave solution. We consider a tapered step function potential V(z)¼ 0 for z o0, V(z)¼maz for 0o zo Dz and V(z)¼ma Dz for z 4 Dz with an incident fast electron of velocity v and described by a plane wave exp(ikz) modulated by a cos(gx/2) factor arising from previous passage through a crystal. As was previously discussed in Section 2.3 for bremsstrahlung emission in crystals, the potential of Eq. (2) can then induce emission of a photon of frequency o. This transition generates decoherence since the gradient operator in Eq. (2) changes the factor cos(gx/2) to sin(gx/2). Momentum conservation is made possible by the presence of the potential step and, as outlined in Appendix B for the simple case of a photon with wave vector q¼ o/c in the z-direction, the probability amplitude can be computed using the WKB method and depends on the integral   Z z Z 1 o o

moVðxÞ dx dz  zþi exp i I¼ ð9Þ 2 v c 1 1 _ k2 v For small values of Dq¼(o/v)  (o/c), this expression is essentially the Fourier transform of an abrupt step function with a result behaving like (Dq)  1. Provided we have infinite path lengths before and after the step, the wave picture seems to confirm the infrared catastrophe described by the classical equation (8) and shows a high probability for decoherence, particularly for the predominant emission close to the beam direction. It also seems very likely that a similar result would be obtained, showing a high decoherence probability for most spontaneous emission events generated in the short segment of magnetic field employed in a magnetic spectrometer, where Eq. (8) also applies. It is clear from both the classical and the wave approach to these examples that the long wave divergence depends on having infinitely long undisturbed trajectories before and after the deceleration event. To control the low frequency divergence in some practical cases therefore, the length of undisturbed path before and after the acceleration region might then provide a more suitable wavelength cut-off than the lateral distance between interfering paths, which was employed above. In electron holography, fringe shifts or decoherence effects can be generated by an applied magnetic field [24] or less controllably by stray fields, whose screening is increasingly crucial in electron optics [25]. Bremsstrahlung emission could also in principle give rise to decoherence [26]. The possible influence of all these effects for the electron holography configuration of Fig. 3 can be checked by another model wave computation. The motion of the two waves diverging from the point S and eventually being brought together again in a distance D to form the interference fringes can be described by the presence of a simple harmonic oscillator potential

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mo20y2/2. This could be the appropriate part of the [y2 x2] potential generated by the quadrupole used in [19]. To get appropriate focusing of a 100 keV electron over the propagation distance DE5 cm, we require o0 ¼ pv/DE1010 Hz. For the electron states in this potential, spontaneous emission (equivalent to bremsstrahlung) will then occur at frequency o0 and the halfcycle dipole emission probability can be readily roughly estimated as before from the Larmor formula. Using for the dipole amplitude A¼ym, half the maximum separation DyE10 mm of the two paths, we find the negligibly small value Prad E4  10  11. Since the emission wavelength l E4D is also well beyond the likely cut-off to cause decoherence, it seems that the effect of bremsstrahlung can be safely dismissed in this case. The influence on electron optical resolution and decoherence of 50 Hz stray electromagnetic fields offers another more familiar and practical example at wavelengths far beyond any cut-off set by the separation of classical paths. For such very longwavelength fields, an electron will suffer correlated phase shifts on both paths and the interference pattern will show an adiabatic time-dependent displacement, which will be averaged over the recording time. A static magnetic field H along the x-direction in Fig. 3 can be represented by adding a vector potential with a y-component Hz in the simple harmonic oscillator wave problem just outlined. The effect of the magnetic field is then to displace the origin of the oscillator potential from y¼0 to y¼yH ¼eH:k/ m2o2 ¼eHv/mo2 ¼eHD/pmo. Cosine interference fringes cos(gx/ 2) at S will now be centred on y¼  yH and after the half cycle propagation in the oscillator potential will appear at y¼ +yH. In a slowly varying field, the consequent averaging effect over the exposure time will completely wash out the fringes if gyH 4 p. Since :g ¼mo Dy/2, this critical condition corresponds to (eH/p:)DDy¼ p making manifest the well known role of the number of magnetic flux quanta in the loop area. Clearly an electric field in the y-direction would produce a similar effect.

5. Incoherent background removal from diffraction patterns Although scattered amplitudes and phases can be recovered from coherent out-of-focus high resolution EM images or from electron holograms, the possibility of phase retrieval simply from diffraction data is exciting. Even where good experimental data about image amplitudes are already to hand, such as in diffraction contrast (including weak beam imaging) of dislocations, it is sometimes insufficient to distinguish between different core structures [27] and the extra phase information could perhaps be helpful. Furthermore, small voids or other point defect clusters with a significant phase contrast component can be difficult or even impossible to detect in amplitude images [28]. For such defects, analysis of the diffraction pattern intensities in the vicinity of Bragg spots already offers a useful point of comparison with X-ray data [29,30] and phase retrieval would be an obvious next step. With sufficient oversampling, phases could be retrieved from energy filtered, single shot diffraction patterns but even using more sophisticated methods such as pytchography with several positions of a focused probe [31], less demanding electron optics would be needed than for electron holography. As noted above, side band holography does however automatically provide elimination of the background from thermal diffuse scattering [8] and some investigation is required to establish whether this can be adequately achieved by other means. Kirk et al. [29] obtained diffraction patterns with partially focused illumination and compared the intensities with the probe on and just off the defect of interest. Unfortunately their assumption that simple subtraction could then be used to eliminate thermal diffuse scattering from the defect data can be easily seen not to be

necessarily valid. From standard diffraction contrast theory we know that the presence of a crystal defect not only transfers intensity from one Bragg spot to the vicinity of another by elastic scattering but also affects the thermal diffuse scattering, particularly if the defect is not too close to the exit surface. Dislocations are after all readily visible in HAADF STEM images. For a known or assumed defect configuration we can of course compute the complete elastic scattering in amplitude and phase allowing for absorption effects. We can also determine the total diff thermal diffuse scattering Itot (integrated over all angles) by subtracting from unity the total computed elastic scattering intensity in all the Bragg spots. In the vicinity of each Bragg spot g diff the diffuse scattering will be a certain fraction fg of Itot . If, as seems reasonable, we assume that the presence or absence of the diff defect simply alters the magnitude of Itot but does not change its angular distribution, neither fg nor the angular shape of the diffuse scattering near g will be changed. Using the symbols p and a to denote presence or absence of the defect, we can then write

Ig diff ðpÞ Ig

diff

¼

ðaÞ

1Sg Ig el ðpÞ 1Sg Ig el ðaÞ

ð10Þ

For a known or assumed defect configuration in a specific crystal we can compute everything on the right of this equation to obtain a correction factor representing the effect of the defect on the local diffuse scattering near g. In the absence of such complete structural knowledge, it may however be adequate to rewrite the equation using on the right hand side experimentally measured Bragg spot intensities Iexp which inevitably include the diffuse g background but restricting the summation to the strongest spots, where the effect of the defect is most pronounced and where the diffuse scattering component is relatively small: Ig diff ðpÞ Ig

diff

ðaÞ

¼

1Sug Ig exp ðpÞ 1Sug Ig exp ðaÞ

ð11Þ

This could be a useful initial step in progress towards a final solution for the defect configuration, which could then be tested using Eq. (10).

6. Conclusions Apart from thermal diffuse scattering and extremely low loss electronic or bremsstrahlung radiation processes, energy filtering can eliminate many sources of decoherence in electron microscopy and diffraction. Despite the low frequency divergence problem, the Larmor formula provides a reasonably reliable means of estimating bremsstrahlung event probabilities. They seem likely to be negligible in most cases except for large-angle bending magnets like mandolin systems or close encounters with large radiating objects. Extremely low energy electronic excitation, by for instance an aloof beam travelling some distance outside a poorly conducting solid, can produce detectable decoherence effects in electron holography.

Acknowledgements I am grateful to Javier Garcia de Abajo and John Spence for valuable discussions.

Appendix A Expressions for the spectral and angular distribution of bremsstrahlung radiation emitted by an electron on a possibly

A. Howie / Ultramicroscopy 111 (2011) 761–767

relativistic trajectory r(t) are given by Jackson [15] but an adequate basis for most of the topics considered here is provided by the simple non-relativistic Larmor equation for power radiated by an electron experiencing an acceleration a(t): dErad e2 ¼ ½aðtÞ2 dt 6pe0 c3

ðA1Þ

To be properly applied this expression has to be integrated over all time or, making use of Parseval’s theorem, the Fourier transform a(o) of the acceleration a(t) has to be integrated over all frequencies:  Z 1  Z 1 e2 e2 2 Erad ¼ ½aðtÞ dt ¼ ½aðoÞ2 do ðA2Þ 6pe0 c3 1 12p2 e0 c3 1 Assuming that the energy loss is due to radiation of single photons, we can get the event probability by confining the second integral to positive frequencies and dividing by :o:  Z 1 e2 1 Prad ¼ ½aðoÞ2 do ðA3Þ 6p2 e0 _c3 0 o Considering first the case of uniform acceleration over a time interval T from an initial velocity v1 to a final velocity v2 we find that Prad exhibits a low frequency logarithmic divergence:   Z 1 e2 sin2 ðoT=2Þ 2 v Þ do ðA4Þ ðv Prad ¼ 2 1 6p2 e0 _c3 oðoT=2Þ2 0 The transition radiation expression of Eq. (4) can be obtained from Eq. (A4) after modifications to allow for the presence of both the charge and its image charge as well as the ability to radiate only into the vacuum half space outside the metal. We next consider dipole radiation at frequencies near O where, but only over a limited time interval T/2ot oT/2, a(t)¼AO2 cos(Ot). With T¼ p/O, only a half cycle occurs and with T¼2p/O, we have one full cycle. The results for Erad and Prad in these two cases are Z 2e2 A2 O3 1 cos2 ðpx=2Þ e2 A2 O3 ¼ dx ¼ Ehalfcycle rad 2 3 2 2 3p e0 c 0 12e0 c3 ð1x Þ 2 2 2 3Z 1 2 2 2 3 2e A O x sin ðpxÞ e A O Ecycle ¼ dx ¼ rad 3p2 e0 c3 0 6e0 c3 ð1x2 Þ2 Z 2 1 2e2 A2 O cos2 ðpx=2Þ e2 A2 O2 halfcycle Prad ¼ dx ¼ Fðx1Þ 2 3 2 3p e0 _c x1 xð1x2 Þ 12e0 _c3 Z 2 2 2 1 2e2 A2 O x sin ðpxÞ e2 A2 O cycle ¼ dx ¼ 0:91 ðA5Þ Prad 3p2 e0 _c3 0 ð1x2 Þ2 6e0 _c3 As would be expected from Parseval’s theorem, the Larmor formula agrees exactly with both of these expressions for radiation energy even if it is no longer confined to a single frequency O. The event probability for the half cycle case shows the same divergence at low frequencies as seen in Eq. (A4) bearing in mind the velocity change v1  v2 ¼2AO. As a function of the lower cut-off frequency o/O ¼x1, function F(x1), denoting a correction to the simple Larmor estimate, rises from 1.04 at x1 ¼0.3 to 2.5 at x1¼0.05. In the full cycle, where v1 ¼v2, there is no similar divergence in the accurate expression for event probability, which differs by less than 10% from the estimate based on Larmor’s formula.

Appendix B An electron deceleration a in a distance Dz can be induced by a potential energy V(z)¼0 for z o0, V(z)¼maz for 0oz o Dz and V(z) ¼ma Dz for z4 Dz. Using the WKB method and ignoring exponential pre-factors, a non-relativistic electron of energy E can

767

then be described by the function  Z z  CðE,zÞ ¼ exp i SðzÞdz cosðgx=2Þ 1

mVðzÞ

SðzÞ  k 2 _ k   _2 g2 2 k þ E¼ 2m 4

ðB1Þ

Here we imagine that the cos(gx/2) image contrast has been written on to the beam by earlier passage through a crystal. Now consider the spontaneous decay to a state with a lower energy E0 ¼E  :o and wave vector k0 ¼k  o/v by emission of a photon with wave vector q¼ o/c in the z-direction. The final state would then be described by a wave function  Z z  CuðEu,zÞ ¼ exp i SuðzÞdz sinðgx=2Þ 1   mVðzÞ o mVðzÞ 1þ 2 ðB2Þ SuðzÞ  ku 2 0  SðzÞ v _ k _ k2 Here the switch from cos(gx/2) to sin(gx/2) and consequent loss of coherence arises through the gradient operator of the radiation potential in Eq. (2). Conservation of momentum in the transition depends on the integral   Z 1 Z z o I¼ exp i z þi ðSa ðxÞSb ðxÞÞdx dz c 1   Z z Z 1 1 o o

moVðxÞ d  zþi ¼ exp i x dz ðB3Þ 2 v c 1 1 _ k2 v

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