Brent Fultz · James Howe Transmission Electron Microscopy and Diffractometry of Materials High resolution transmission electron microscope (HRTEM) image of a lead crystal between two crystals of aluminum (i.e., a Pb precipitate at a grain boundary in Al) The two crystals of Al have different orientations, evident from their different patterns of atom columns Note the commensurate atom matching of the Pb crystal with the Al crystal at right, and incommensurate atom matching at left An isolated Pb precipitate is seen to the right The HRTEM method is the topic of Chapter 10 Image courtesy of U Dahmen, National Center for Electron Microscopy, Berkeley Brent Fultz · James Howe Transmission Electron Microscopy and Diffractometry of Materials Third Edition With 440 Figures and Numerous Exercises 123 Prof Dr Brent Fultz California Institute of Technology Materials Science and Applied Physics MC 138-78 Pasadena CA 91125 USA btf@caltech.edu http://www.its.caltech.edu/ matsci/btf/Fultz1.html Prof Dr James M Howe University of Virginia Department of Materials Science and Engineering P O Box 400745 Charlottesville VA 22904-4745 USA jh9s@virginia.edu http://www.virginia.edu/ms/faculty/howe.html Library of Congress Control Number: 2007933070 ISSN: 1439-2674 ISBN 978-3-540-73885-5 3rd Edition Springer Berlin Heidelberg New York ISBN 978-3-540-43764-2 2nd Edition Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2001, 2002, 2008 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: supplied by the authors Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover Design: eStudioCalamar S.L., F Steinen-Broo, Girona, Spain SPIN 12085156 57/3180/YL 543210 Printed on acid-free paper Preface Experimental methods for diffraction and microscopy are pushing the front edge of nanoscience and materials science, and important new developments are covered in this third edition For transmission electron microscopy, a remarkable recent development has been a practical corrector for the spherical aberration of the objective lens Image resolution below ˚ A can be achieved regularly now, and the energy resolution of electron spectrometry has also improved dramatically Locating and identifying individual atoms inside materials has been transformed from a dream of fifty years into experimental methods of today The entire field of x-ray spectrometry and diffractometry has benefited from advances in semiconductor detector technology, and a large community of scientists are now regular users of synchrotron x-ray facilities The development of powerful new sources of neutrons is elevating the field of neutron scattering research Increasingly, the most modern instrumentation for materials research with beams of x-rays, neutrons, and electrons is becoming available through an international science infrastructure of user facilities that grant access on the basis of scientific merit The fundamentals of scattering, diffractometry and microscopy remain as durable as ever This third edition continues to emphasize the common theme of how waves and wavefunctions interact with matter, while highlighting the special features of x-rays, electrons, and neutrons The third edition is not substantially longer than the second, but all chapters were updated and revised The text was edited throughout for clarity, often minimizing sources of confusion that were found by classroom teaching There are significant changes in Chapters 1, 3, 7, and Chapter 11 is new, so there are now 12 chapters in this third edition Many chapter problems have been tuned to minimize ambiguity, and the on-line solutions manual has been updated We thank Drs P Rez and A Minor for their advice on the new content of this third edition Both authors acknowledge support from the National Science Foundation for research and teaching of scattering, diffractometry, and microscopy Brent Fultz and James Howe Pasadena and Charlottesville May, 2007 VI Preface to the First Edition Preface to the Second Edition We are delighted by the publication of this second edition by Springer–Verlag, now in its second printing The first edition took over twelve years to complete, but its favorable acceptance and quick sales prompted us to prepare the second edition in about two years The new edition features many re-writings of explanations to improve clarity, ranging from substantial re-structuring to subtle re-wording Explanations of modern techniques such as Z-contrast imaging have been updated, and errors in text and figures have been corrected over the course of several critical re-readings The on-line solutions manual has been updated too The first edition arrived at a time of great international excitement in nanostructured materials and devices, and this excitement continues to grow The second edition shows better how nanostructures offer new opportunities for transmission electron microscopy and diffractometry of materials Nevertheless, the topics and structure of the first edition remain intact The aims and scope of the book remain the same, as our teaching suggestions We thank our physics editors Drs Claus Ascheron and Angela Lahee, and our production editor Petra Treiber of Springer–Verlag for their help with both editions Finally, we thank the National Science Foundation for support of our research efforts in microscopy and diffraction Brent Fultz and James Howe Pasadena and Charlottesville September, 2004 Preface to the First Edition Aims and Scope of the Book Materials are important to mankind because of their properties such as electrical conductivity, strength, magnetization, toughness, chemical reactivity, and numerous others All these properties originate with the internal structures of materials Structural features of materials include their types of atoms, the local configurations of the atoms, and arrangements of these configurations into microstructures The characterization of structures on all these spatial scales is often best performed by transmission electron microscopy and diffractometry, which are growing in importance to materials engineering and technology Likewise, the internal structures of materials are the foundation for the science of materials Much of materials science has been built on results from transmission electron microscopy and diffractometry of materials This textbook was written for advanced undergraduate students and beginning graduate students with backgrounds in physical science Its goal is to acquaint them, as quickly as possible, with the central concepts and some details of transmission electron microscopy (TEM) and x-ray diffractometry Preface to the First Edition VII (XRD) that are important for the characterization of materials The topics in this book are developed to a level appropriate for most modern materials characterization research using TEM and XRD The content of this book has also been chosen to provide a fundamental background for transitions to more specialized techniques of research, or to related techniques such as neutron diffractometry The book includes many practical details and examples, but it does not cover some topics important for laboratory work such as specimen preparation methods for TEM Beneath the details of principle and practice lies a larger goal of unifying the concepts common to both TEM and XRD Coherence and wave interference are conceptually similar for both x-ray waves and electron wavefunctions In probing the structure of materials, periodic waves and wavefunctions share concepts of the reciprocal lattice, crystallography, and effects of disorder Xray generation by inelastic electron scattering is another theme common to both TEM and XRD Besides efficiency in teaching, a further benefit of an integrated treatment is breadth – it builds strength to apply Fourier transforms and convolutions to examples from both TEM and XRD The book follows a trend at research universities away from courses focused on one experimental technique, towards more general courses on materials characterization The methods of TEM and XRD are based on how wave radiations interact with individual atoms and with groups of atoms A textbook must elucidate these interactions, even if they have been known for many years Figure 1.12, for example, presents Moseley’s data from 1914 because this figure is a handy reference today On the other hand, high-resolution TEM (HRTEM), modern synchrotron sources, and spallation neutron sources offer new ways for wavematter interactions to probe the structures of materials A textbook must integrate both these classical and modern topics The content is a confluence of the old and the new, from both materials science and physics Content The first two chapters provide a general description of diffraction, imaging, and instrumentation for XRD and TEM This is followed in Chapters and by electron and x-ray interactions with atoms The atomic form factor for elastic scattering, and especially the cross sections for inelastic electron scattering, are covered with more depth than needed to understand Chapters 5–7, which emphasize diffraction, crystallography, and diffraction contrast In a course oriented towards diffraction and microscopy, it is possible to take an easier path through only Sects 3.1, 3.2.1, 3.2.3, 3.3.2, and the subsection in 3.3.3 on Thomas–Fermi and Rutherford models Similarly, much of Sect 4.4 on core excitations could be deferred for advanced study The core of the book develops kinematical diffraction theory in the Laue formulation to treat diffraction phenomena from crystalline materials with increasing amounts of disorder The phase-amplitude diagram is used heavily in Chapter for the analysis of diffraction contrast in TEM images of defects After a treatment of diffraction lineshapes in Chapter 8, the Patterson function is used in Chapter to treat short-range order phenomena, VIII Preface to the First Edition thermal diffuse scattering, and amorphous materials High-resolution TEM imaging and image simulation follow in Chapter 10, and the essentials of the dynamical theory of electron diffraction are presented in Chapter 11 [now 12 in the third edition] With a discussion of the effective extinction length and the effective deviation parameter from dynamical diffraction, we extend the kinematical theory as far as it can go for electron diffraction We believe this approach is the right one for a textbook because kinematical theory provides a clean consistency between diffraction and the structure of materials The phase-amplitude diagram, for example, is a practical device for interpreting defect contrast, and is a handy conceptual tool even when working in the laboratory or sketching on table napkins Furthermore, expertise with Fourier transforms is valuable outside the fields of diffraction and microscopy Although Fourier transforms are mentioned in Chapter and used in Chapter 3, their manipulations become more serious in Chapters 4, and Chapter presents convolutions, and the Patterson function is presented in Chapter The student is advised to become comfortable with Fourier transforms at this level before reading Chapters 10 and 11 [now 10-12 in the third edition] on HRTEM and dynamical theory The mathematical level is necessarily higher for HRTEM and dynamical theory, which are grounded in the quantum mechanics of the electron wavefunction Teaching This textbook evolved from a set of notes for the one-quarter course MS/APh 122 Diffraction Theory and Applications, offered to graduate students and advanced undergraduates at the California Institute of Technology, and notes for the one-semester graduate courses MSE 703 Transmission Electron Microscopy and MSE 706 Advanced TEM, at the University of Virginia Most of the students in these courses were specializing in materials science or applied physics, and had some background in elementary crystallography and wave mechanics For a one-semester course (14 weeks) on introductory TEM, one of the authors covers the sections: 1.1, 2.1–2.8, 3.1, 3.3, 4.1-4.3, 4.6, 5.1–5.6, 6.1-6.3, 7.1–7.14 In a course for graduate students with a strong physics background, the other author has covered the full book in 10 weeks by deleting about half of the “specialized” topics [He has never repeated this achievement, however, and typically manages to just touch section 10.3.] The choice of topics, depth, and speed of coverage are matters for the taste and discretion of the instructor, of course To help with the selection of course content, the authors have indicated with an asterisk, “*,” those sections of a more specialized nature The double dagger, “‡,” warns of sections containing a higher level of mathematics, physics, or crystallography Each chapter includes several, sometimes many, problems to illustrate principles The text for some of these problems includes explanations of phenomena that seemed too specialized for inclusion in the text itself Hints are given for some of the Preface to the First Edition IX problems, and worked solutions are available to course instructors Exercises for an introductory laboratory course are presented in an Appendix When choosing the level of presentation for a concept, the authors faced the conflict of balancing rigor and thoroughness against clarity and conciseness Our general guideline was to avoid direct citations of rules, but instead to provide explanations of the underlying physical concepts The mathematical derivations are usually presented in steps of equal height, and we try to highlight the central tricks even if this means reviewing elementary concepts The authors are indebted to our former students for identifying explanations and calculations that needed clarification or correction Acknowledgements We are grateful for the advice and comments of Drs C C Ahn, D H Pearson, H Frase, U Kriplani, N R Good, C E Krill, Profs L Anthony, L Nagel, M Sarikaya, and the help of P S Albertson with manuscript preparation N R Good and J Graetz performed much of the mathematical typesetting, and we are indebted to them for their careful work Prof P Rez suggested an approach to treat dynamical diffraction in a unified manner Both authors acknowledge the National Science Foundation for financial support over the years Brent Fultz and James Howe Pasadena and Charlottesville October, 2000 Contents Diffraction and the X-Ray Powder Diffractometer 1.1 Diffraction 1.1.1 Introduction to Diffraction 1.1.2 Bragg’s Law 1.1.3 Strain Effects 1.1.4 Size Effects 1.1.5 A Symmetry Consideration 1.1.6 Momentum and Energy 1.1.7 Experimental Methods 1.2 The Creation of X-Rays 1.2.1 Bremsstrahlung 1.2.2 Characteristic Radiation 1.2.3 Synchrotron Radiation 1.3 The X-Ray Powder Diffractometer 1.3.1 Practice of X-Ray Generation 1.3.2 Goniometer for Powder Diffraction 1.3.3 Monochromators, Filters, Mirrors 1.4 X-Ray Detectors for XRD and TEM 1.4.1 Detector Principles 1.4.2 Position-Sensitive Detectors 1.4.3 Charge Sensitive Preamplifier 1.4.4 Other Electronics 1.5 Experimental X-Ray Powder Diffraction Data 1.5.1 * Intensities of Powder Diffraction Peaks 1.5.2 Phase Fraction Measurement 1.5.3 Lattice Parameter Measurement 1.5.4 * Refinement Methods for Powder Diffraction Data Further Reading Problems 1 10 10 13 14 16 20 23 23 25 28 30 30 34 36 37 38 38 45 49 52 54 55 The TEM and its Optics 2.1 Introduction to the Transmission Electron Microscope 2.2 Working with Lenses and Ray Diagrams 2.2.1 Single Lenses 61 61 66 66 532 10 High-Resolution TEM Imaging 10.2 Physical Optics of High-Resolution Imaging This section develops a set of mathematical tools that are useful for calculating contrast in high-resolution images Different mathematical functions correspond to wave propagation, lenses, and even materials The mathematical operations are primarily Fourier transforms and convolutions of Gaussian functions and delta functions Similar manipulations were performed in Sect 9.4.2, and were first discussed in Sect 8.2 In essence, an optical model with components of propagating wavefronts (pR ), specimens (qi ), and lenses (qlens ) is converted to a mathematical model of products or convolutions of real-space functions (q and p) or their Fourier transforms (Q and P ) Each function corresponds to a component of the model The choice of a real-space function or a k-space function is usually made for the purpose of replacing an awkward convolution of two functions with a more convenient multiplication of their Fourier transforms The presentation of the Huygens principle in the previous Sect 10.1 motivates the definition of a wavefront propagator, which is a kernel of the Green’s function of the wave equation This propagator, pR , expands a spherical wave outwards over the distance R A lens function, qlens , provides the opposite action, and has the mathematical form to converge a plane wave into a point over the distance of one focal length, f The specimen function, qi , discussed in Sect 10.2.3, provides phase shifts (and also absorption) to the wave front The set of mathematical tools presented in this Sect 10.2 is well-suited for understanding the effects of lens defects on high-resolution TEM images 10.2.1 ‡ Wavefronts and Fresnel Propagator In Sect 10.1.2, all points on the surface of a spherical wavefront at r0 were assumed to be point emitters of spherical waves This implementation of the Huygens principle predicted the correct forward propagation of the spherical wave The actual work involved performing a convolution of the spherical wave propagator with the incident wavefront It was essentially the procedure for solving the Schră odinger wave equation with the method of Green’s functions In both cases the Green’s function “kernel,” (10.7) or (10.38) below, is the spherical wave emitted by a single point on the wavefront To calculate the total scattered wave, this point response was convoluted with the amplitude over the entire wavefront, (10.6) or (10.12) Here we define the Green’s function kernel, or “propagator” (of spherical waves), as: −i ikR e (10.38) Rλ This p(R), convoluted with the surface of the wavefront in (10.12), provides the scattered wave amplitude at point P Since R2 = x2 + y + z : p(R) ≡ 10.2 Physical Optics of High-Resolution Imaging 533 −i ik(x2 +y2 +z2 )/R e (10.39) Rλ The factor 1/λ is necessary to obtain the correct intensity when integrating over Fresnel zones as in Fig 10.7 As explained following (10.24), waves with larger λ and smaller k have wider phase-amplitude spirals, and would have larger amplitudes unless we normalized by λ The factor of −i compensates for the phase shift of the Fresnel integral, as explained after (10.24) , assume small angles of We now put the propagation direction along z scattering so that z R, and therefore ignore the z-dependence of p(x, y, z) in (10.39).7 We work instead with the “Fresnel propagator,” pR (x, y): p(x, y, z) = −i ik(x2 +y2 )/R e (10.40) Rλ This propagator is convoluted with a wavefront to move the wavefront forward by the distance R As a first example, we apply the propagator to an incident spherical wavefront Section 10.1.2 worked the details of this convolution of the propagator with a spherical wavefront, qsphr (x, y): pR (x, y) = ik(x2 +y2 )/r e , r so from (10.24) we know the result: qsphr (x, y) = (10.41) 2 eik(x +y )/(R+r) (10.42) R+r Anticipating the multislice method of Sect 10.2.3, we use the notation Ψi (x, y) for the incident wave, and Ψi+1 (x, y) for the wave after the operation of the propagator In another example of the use of the Fresnel propagator, consider the wave emitted by a point source, qδ (x, y), which is a product of two Dirac delta functions: Ψi+1 (x, y) = qsphr (x, y) ∗ pR (x, y) = qδ (x, y) = δ(x) δ(y) (10.43) The variables x and y are independent, so the convolution with of (10.40) each delta function of (10.43) simply returns exp ikx2 /R and exp iky /R : Ψi+1 (x, y) = qδ (x, y) ∗ pR (x, y) , i ik(x2 +y2 )/R i ik(x2 +y2 )/R e e = Ψi+1 (x, y) = δ(x) δ(y) ∗ Rλ Rλ The intensity is: ∗ Ψi+1 = Ψi+1 λ2 R2 (10.44) (10.45) (10.46) Note that exp(ikz /R) exp(ikR), which has no effect on the intensity because exp(ikR) exp(−ikR) = 534 10 High-Resolution TEM Imaging The point source wavefront of (10.43), convoluted with the propagator, gives a wave intensity that decreases as R−2 , as expected for a spherical wave The factor of λ−2 was not obtained in the correct (10.42), however, even as we let the r in (10.41) go to zero More deftness is required in performing the delta function convolutions than we used in (10.45) In most of what follows, however, we simply ignore the prefactor for the Fresnel propagator, and avoid the trouble of taking the delta function as a limit of a small spherical wavefront 10.2.2 ‡ Lenses Figure 2.34 showed the essence of how to design a lens by considering phase shifts, and this concept is also shown in the center of Fig 10.4 in the context of the Huygens principle This section presents the lens as a mathematical phase shifter The lens is considered to be a planar object, providing phase shifts across an x-y plane An ideal lens of focal length f has the phase function: qlens (x, y) = e−ik(x +y )/f (10.47) The lens distorts the phases of a wavefront at its location, so the wavefront is multiplied by qlens (x, y) at the position of the lens Note that the phase itself increases parabolically from the optic axis (as x2 + y in (10.47)), consistent with (2.23) and our assumption of paraxial rays Rules The rules for working with lenses and propagators are: • Lenses (and materials), denoted “q(x, y),” are assumed infinitesimally thin, and their action is to make phase shifts in a wavefront These objects multiply the wavefront at their locations in real space (Lens distortions, however, are best parameterized k-space, where lens and material functions, Q(Δkx , Δky ), must be convoluted rather than multiplied.) A • Propagators, denoted “p(x, y),” move the wavefront forward along z single point is propagated as a spherical wave, but the full wavefront must be convoluted with p(x, y) to move it forward (When the wavefront can be expressed as a set of diffracted beams in k-space, the propagator, P (Δkx , Δky ), operates on the wavefront by multiplication rather than convolution.) Example One Consider a plane wave that passes through a lens, and propagates a distance f , where f is the focal length of the lens We know that the wave, Ψi+1 (x, y), must be focused to a point after these operations The final wave is:8 Note the alternative k-space formulation of (10.48): Ψi+1 (Δk) = Ψi (Δk) ∗ Qlens (Δk) Pf (Δk) 10.2 Physical Optics of High-Resolution Imaging Ψi+1 (x, y) = Ψi (x, y) qlens (x, y) ∗ pf (x, y) 535 (10.48) For simplicity we ignore the prefactors in (10.40), and work with the xdimension only The wavefront of a plane wave has no variation with x, so we represent it as the factor With (10.40) and (10.47), (10.48) becomes: 2 ψi+1 (x) = e−ikx /f ∗ eikx /f (10.49) Section 8.1.3 (8.23) noted that the convolution of two Gaussians is another Gaussian The breadths add in quadrature, even if they are complex numbers For (10.49) we find a breadth, σ: f f + =0 (10.50) σ= −ik ik A Gaussian of zero breadth is a delta function, so (10.49) becomes: ψi+1 (x) = δ(x) (10.51) The function for the ideal lens (10.47) causes, as expected, a plane wave passing through the lens to be focused to a point at the distance f Example Two Consider a point source of illumination, propagated a distance d2 to the lens, passed through the lens, and propagated to a focal point at the distance d1 on the other side of the lens This is the situation shown in Fig 2.32 Our formalism for propagators and lens becomes9 : ψi+1 (x, y) = qδ (x, y) ∗ pd2(x, y) qlens (x, y) ∗ pd1(x, y) (10.52) For simplicity, we work with one dimension only (x), and ignore the prefactor for the propagator in (10.40) Equation (10.52) becomes: 2 ψi+1 (x) = δ(x) ∗ eikx /d2 e−ikx /f ∗ eikx /d1 (10.53) We know from the lens formula (2.1) that for a point source to be focused to a point, the distance of propagation from the left and right are related as: 1 = − , (10.54) d2 f d1 so when we substitute (10.54) into (10.53), 2 ψi+1 (x) = δ(x) ∗ eikx (1/f −1/d1 ) e−ikx /f ∗ eikx /d1 , ψi+1 (x) = δ(x) ∗ e−ikx /d1 ∗ eikx /d1 (10.55) (10.56) As discussed for (10.49) and (10.50), the second convolution is δ(x), so: ψi+1 (x) = δ(x) (10.57) This second example showed how we can use phase shifts by lenses with propagators to take a point source of illumination through a lens and focus it to a point, given that the lens formula is satisfied Note the alternative k-space formulation of (10.52): Ψi+1 (Δk) Ψi (Δk) Pd2(Δk) ∗ Qlens (Δk) Pd1(Δk) For our point source, Ψi (Δk) = = 536 10 High-Resolution TEM Imaging Lens Distortions The present formalism will be used in Sect 10.3.2 for the analysis of non-ideal lenses Lens defects modify the phase shift of the lens, and are included as a factor that multiplies the lens transfer function in k-space The essential features of this phase transfer function, exp iW(Δk) , are presented in k-space in Sect 10.3.3 To work with the lens function of (10.47) transform of in real space, however, we convolute it with the Fourier (x, y): exp iW(Δk) to obtain the performance of a real lens, qlens 2 (10.58) qlens (x, y) = e−ik(x +y )/f ∗ F e−iW(Δk) In (10.58) we have written the phase transfer function as a function of Δk, which involves the angle made by an electron with respect to the optic axis as it enters the lens Ideal lens performance is possible only if W(Δk) is a constant.10 We expect, however, that spherical aberration will cause W(Δk) to increase with Δk, and we evaluate this problem in detail in Sects 10.3.1 (x, y) by adjusting f to 10.3.3, with emphasis on how to optimize qlens 10.2.3 ‡ Materials The present “physical optics approach” of wave propagators, wavefronts, and phase transfer functions of lenses is well-suited for computer simulations of high-resolution TEM images, as developed in Sect 10.4 Consider the general through N layers of mateexpression for the electron wave traveling along z rial Each layer advances the phase of the wavefront by small amounts, and these amounts differ at various x, y over the layer (corresponding to atomic columns and channels) This phase advance through the layer is given by the multiplicative factor, qi (x, y), or symbolically, qi (x) or qi (A layer of empty space has qi (x) = 1.) We have to convolute this new wavefront after the layer with a propagator pi (x) to move the wavefront to the next layer The following expression for the wave just modified by the N th layer of material is simple if you first look at the zeroth layer in the center of the equation, using numbers below the brackets to match them in pairs: ψN +1 (x) = qN (x) q2 q1 q0 ∗ p0 ∗ p1 ∗ p2 ∗ pN −1 (x) N 1 N −1 N (10.59) In its alternative formulation in Fourier space, where Q(Δk) ≡ F −1 [q(x)] and P (Δk) ≡ F −1 [p(x)], this equation involves multiplications of the propagators instead of convolutions: ψN +1 (Δk) = QN (Δk) ∗ Q2 ∗ Q1 ∗ Q0 P0 P1 P2 PN −1 (Δk) N 1 N −1 N (10.60) 10 In this case, exp(−iW(Δk)) is a constant of modulus 1, so its Fourier transform is a δ-function The convolution in (10.58) of this δ-function with the ideal lens function, exp{−ik[(x2 + y )/f ]}, returns the ideal lens function 10.2 Physical Optics of High-Resolution Imaging 537 The propagators, pi (x), are assumed to be the same as in (10.40) In other words, the electron wavefront propagates between layers as if in a vacuum The layers themselves are assumed infinitesimally thin, and provide only a phase shift, qi (x), and no propagation We know the form of the free space propagators, but what is the meaning of qi (x) for the material? In general, qi (x, y) has the form: qi (x, y) = e−iσ φi(x,y)−μ(x,y) (10.61) The first term in the exponent provides for a phase shift that varies with position, (x, y), and the second term provides for absorption It is the role of the dynamical theory of diraction to calculate q, starting with the Schră odinger equation, and some aspects of a crystal as a “phase grating” are presented in Sect 11.2.3 We can relate the phase distortion to the effective potential of the electron in the material To so, we make use of the fact that the electron wavevector in the crystal, k, differs from the wavevector in the vacuum, χ, because the potential energy for the electron in the crystal is −eV (the potential is attractive because the electron passes through positive ion cores) To conserve total energy, the kinetic energy of the electron in the crystal must increase by +eV to compensate for the potential energy, so while the wavevector in vacuum, χ, is: 2mE0 , (10.62) χ= 2 the wavevector in the crystal, k, is slightly larger: 2m (E0 + eV ) k= , (10.63) 2 eV 2mE0 1+ , (10.64) k 2E0 eV k χ 1+ (10.65) 2E0 At a snapshot in time at t , the wave ψ(kz − ωt ) has a phase, kz − ωt , that increases by the amount k dz over the distance interval from z to z + dz Over this small distance interval, the plane wave ψz = exp(ikz) changes into ψz+dz = exp ik(z + dz) = ψz exp(ik dz) After propagating in a material of average potential −eV from z to z + dz, the k of (10.65) gives the plane wave: eV dz (10.66) ψz+dz ψz eiχ dz exp ik 2E0 The first exponential is as expected when the electron propagates through vacuum (cf., (10.38)) The second exponential in (10.66) is more interesting because V depends on position (since atoms are located at various x, y, z) The potential V is not homogeneous in x, y when atoms lie along columns, and we are interested in how electrons traveling down columns at different 538 10 High-Resolution TEM Imaging x, y experience different V After a plane wave has propagated the thickness t, the new wavefront is found by summing (integrating) all phase shifts in the exponents of (10.66): ψz+t = ψz eiχt exp ike t V (x, y, z) dz 2E0 (10.67) The multislice calculational scheme of (10.59) and (10.60) assumes these phase shifts occur in layers infinitesimally thin, but spaced apart by the distance t The phase shift and absorption of the infinitesimal layer is equal to that caused by a thickness, t, of material The nth layer multiplies the wavefront by qn , where: ike t qn (x, y) = exp V (x, y, z) dz 2E0 (10.68) Using this qn in (10.59) to represent the effect of a thin layer of material, the propagator of (10.40) then moves the wavefront by the distance, t, to the next layer The choice of thickness, t, is discussed further in Sect 10.4 Certainly this type of wave scattering calculation is accurate when t is subatomic, but much larger values of t (some fraction of the extinction distance) are acceptable in practice To make further progress we need a “multislice” computer calculation code as described in Sect 10.4 In principle, these calculations use expressions such as (10.68) for q and (10.40) for p The multislice computer code performs a series of operations as in (10.59) and (10.60), where the phase distortion of a wave incident on the ith layer is calculated as a function of x and y, the th wave is propagated to the (i + 1) layer, and the process is repeated Before we return to these issues in more detail, however, we next describe how the objective lens alters the phase of the electron wavefront 10.3 Experimental High-Resolution Imaging 10.3.1 Defocus and Spherical Aberration The performance of the objective lens is the central issue in the method of HRTEM We show in Sect 10.3.2 that contrast in high-resolution images originates primarily with the phase shifts of the electron wavefront as it passes through the specimen The objective lens is therefore best understood as a device that alters the phase of the electron wavefront To focus the wavefront, Figs 2.34 and 10.4 show that the phases of the off-axis rays must be advanced with respect to the on-axis ray The phase advance must be done with great precision if the phase-contrast image is to provide meaningful information Conspiring against this precision is the positive third-order 10.3 Experimental High-Resolution Imaging 539 spherical aberration of magnetic lenses (Sect 2.7.1) A positive coefficient of spherical aberration, Cs , means that rays at larger angles to the optic axis will focus closer to the lens (see Fig 2.38) The closer focus means that these off-axis rays have undergone an excessive amount of phase advance by the lens It is unfortunate that all short solenoid magnetic lenses have a positive Cs , especially when they have a large bore and pole-piece gap It is possible, however, to compensate in part for the errors caused by spherical aberration by adjusting the focus of the lens Doing so optimizes the range of angles for which entering rays suffer acceptable phase distortions The larger this range of angles, the larger the usable range of Δk for electrons diffracted from the sample High values of Δk correspond to small distances in real space, so the image has better spatial resolution The compensation of spherical aberration by defocus is not perfect, however, because defocus and spherical aberration depend differently on Δk Optimizing the compensation provides the resolution limit of the microscope, a limit that is achieved regularly when skilled microscopists examine good specimens on well-maintained instruments Effect of Defocus The electron is assumed to originate from a point on the optic axis, and is assumed to make small angles with respect to the optic axis These assumptions are good because the region examined is very small, and the diffraction angles are small too We first calculate errors in bending angle, ε, as a function of R, the radius at which the ray enters the lens Figure 10.12 shows the geometry for the error, εa , in bending angle caused by defocus From the figure, the angle θ is: R (10.69) θ = b Fig 10.12 The error in bend angle, εa , caused by defocus, Δf , is proportional to R, where R is the distance along the radius of the thin lens The ratio of defocus error εa to the angle θ is the same as the ratio of the distance Δb to the distance b, so: Δb θ b Substituting (10.69) in (10.70): εa = (10.70) Δb R (10.71) b2 We need to express εa in terms of the actual defocus, Δf , at the specimen on the left side of the lens in Fig 10.12 Recall the lens formula, (2.1): εa = 540 10 High-Resolution TEM Imaging 1 = + f a b (10.72) For small differences in the lengths a and b (here Δa < and Δb > 0), the lens formula is: 1 = + , (10.73) f a + Δa b + Δb 1 Δa Δb 1− + 1− , (10.74) f a a b b Δa Δb − + − (10.75) f a a b b Substituting (10.72) into (10.75), we obtain: Δb Δa − b2 a We substitute (10.76) into (10.71) for our error in angle: (10.76) Δa R (10.77) a2 The objective lens is operated for high magnification, so b a, and a f from (10.72) The distance, Δa, is the defocus, Δf , so (10.77) becomes: εa = − εa = − Δf R f2 (10.78) Fig 10.13 The error in bend angle caused by spherical aberration, εs , is proportional to R3 (see text) Effect of Third-Order Spherical Aberration Figure 10.13 shows the geometry for the error in bending angle caused by spherical aberration, εs A perfect lens would focus the off-axis rays along the solid line, but positive spherical aberration causes the ray to follow the path of the dashed line.11 From Fig 10.13, the angles θ and εs are: R , (10.79) θ= a Δr εs = (10.80) b 11 By comparing Figs 10.12 and 10.13 we can see immediately how defocus can be used to compensate for spherical aberration, at least for the one ray path at R 10.3 Experimental High-Resolution Imaging 541 The distance, Δr, is proportional to both the spherical aberration, through a factor Cs θ3 , and the magnification, which is b/a: b a Substituting (10.81) into (10.80): Δr = Cs θ3 (10.81) Cs θ3 b/a (10.82) b Using (10.79) for θ, and the approximation at high magnification that a f , (10.82) becomes: εs = εs = Cs R3 f4 (10.83) Compensate Errors of Spherical Aberration by Defocus The errors in bending angle caused by defocus, εa , and spherical aberration, εs , add to give a total error in bending angle, ε: ε = εs + εa , (10.84) and we substitute for εs and εa from (10.78) and (10.83): ε = Cs R3 R − Δf f f (10.85) We will show that this error in angle of bend is proportional to an error in phase First, however, note from Fig 10.14 that for proper focusing, the lens must bend the ray by the angle θ +θ Another ray arriving at the lens further from the optic axis at the distance R + dR must bend more if it is to come to focus With spherical aberration, however, the ray at the position R + dR is bent a bit too much This excess is shown as the angle ε in Fig 10.14 The excessive amount of path length traveled by the ray, dS, over the distance dR is: dS = ε dR (10.86) The error in phase, dW , contributed over the radius dR at R, is therefore: 2π ε dR (10.87) λ The total error in phase is obtained by integrating the contributions, dW , over all R To the integral, we need a reference phase that serves as the lower limit of integration We assign zero phase to the ray along the optic axis Integration of (10.87) is then performed from the center of the lens to R: R 2π W (R) = ε dR , (10.88) λ dW = 542 10 High-Resolution TEM Imaging Fig 10.14 Geometry of the excessive angle of bend, ε, for a lens with positive spherical aberration and using (10.85) as the integrand: R2 2π R4 Cs − Δf W (R) = λ f f (10.89) For high magnification: θ R , f (10.90) so: π Cs θ4 − Δf θ2 (10.91) 2λ The phase shift error is a function of the diffraction vector, Δk, since Δk = 4πθB λ−1 = 2kθB (see Fig 5.4) The θ in W (θ) corresponds to twice the Bragg angle, θB , so for small θ = Δk/k: 4 2 Δk Δk k Cs − 2Δf (10.92) W (Δk) = k k W (θ) = An electron wavelet traveling parallel to k0 + Δk undergoes a phase shift of W (Δk) when it comes to focus in a TEM image Consider first the hypothetical case when W = for all Δk This is ideal for atomic resolution imaging For spatial scales larger than atomic separations, however, the amplitudes of all scattered waves add in phase with the forward beam, so the image is indistinguishable from the case where no scattering occurs When W = 0, which is reasonably achievable at small Δk, diffraction contrast in the image is weak.12 It is therefore useful to enhance the diffraction contrast by using an objective aperture as in bright-field or dark-field imaging 12 When the scattering is incoherent or inelastic (both can be parameterized as “absorption”), some image contrast is expected when W = 0, however 10.3 Experimental High-Resolution Imaging 543 High-resolution TEM requires that Δk be as large as possible, so it is important to understand image contrast in realistic cases where W (Δk) is not small The waves diffracted by the various Δk must have their phase multiplied by a phase transfer function of the objective lens, QPTF (Δk): QPTF (Δk) = e−iW(Δk) (10.93) Since this function QPTF (Δk) is in k-space, and our specimen function qi (x, y) of (10.61) is in real space, we should either transform (10.93) into real space, or transform the specimen function qi (x, y) into k-space Our interest is in how the lens alters the contrast from various periodicities of the sample, so we take the k-space approach 10.3.2 ‡ Lenses and Specimens Lattice Fringe Imaging A simple example shows how the phase transfer function of the objective lens, QPTF (Δk) of (10.93), affects a high resolution image Here the electron wavefunction through the specimen is represented with only the forward beam and one diffracted beam High-resolution imaging is phase coherent imaging, so we add the amplitudes of the two beams: Δz i(k0 +g)·r iW(g) ik0 ·r iW(0) e +i e e (10.94) ψtot = φ0 e ξg The phases exp(ik0 · r) and exp(i (k0 + g)·r) of the forward and diffracted beams, respectively, are altered by the QPTF (Δk) of the objective lens These forward and diffracted beams have specific Δk, so the phase alterations by the lens are W (0) and W (g) Note that W (0) ≡ 0, so exp(iW (0)) = for the forward beam The constant prefactor of the diffracted beam, iφ0 Δz/ξg , is derived in Chapter 11 It includes the incident wave amplitude, φ0 , times the scattering strength of an increment of material The scattering strength naturally depends on the ratio of thickness, Δz, to extinction length, ξg , and we assume Δz ξg The intensity of the electron wavefunction at the image ∗ ψtot : is, as usual, ψtot Δz −i(k0 +g)·r −iW(g) ∗ −ik0 ·r Itot = φ0 e −i e e ξg Δz i(k0 +g)·r iW(g) ik0 ·r × φ0 e +i e e , (10.95) ξg Δz ig·r iW(g) Itot = |φ0 | + i e e ξg Δz −ig·r −iW(g) Δz 2 (10.96) −i e e + ξg ξg We have already assumed that the sample is very thin and the scattering is weak The last term in (10.96), which is of second order in the scattering, can therefore be neglected: 544 10 High-Resolution TEM Imaging Itot = |φ0 | 2Δz sin g·r + W (g) , 1− ξg Itot = |φ0 | − |φ0 | 2Δz sin(g·r) cos W (g) ξg + cos(g·r) sin W (g) (10.97) (10.98) The larger first term in (10.98) is from the forward beam The second term is proportional to the scattering, 1/ξg , and predicts contrast known as “lattice fringes.” These fringes lie perpendicular to g, and have a periodicity 2π/g Both the sin(g·r) and cos(g·r) terms provide fringes of the same periodicity, but with displaced positions on the image The precise position of the observed fringes depends on the phase error, W (g), for the diffracted beam For an image obtained with no defocus (Δf = 0), and a small g for the diffracted beam (small Δk), from (10.92) this phase error is expected to be near zero, so the sin(g·r) term in braces in (10.98) would dominate On the other hand, as discussed below, the best resolution of the microscope is often obtained when W (g) is approximately −π/2, so the cos(g·r) term often dominates in a high-resolution image When only one set of fringes is visible in an image, it is rarely important to know exactly where the fringes are positioned On the other hand, an image showing only one set of fringes is not very informative about the atomic structure of the sample, since this information can be obtained from a diffraction pattern (at least when the crystal is large) A more substantial HRTEM research project may seek the interface structure when two crystals are in physical contact with near-atomic registry Suppose it is possible to obtain lattice fringe images from both crystals, and suppose further that the fringes from both crystals touch each other It might be tempting to claim from inspection of the image that the atomic planes are in alignment across the interface Such an interpretation could be naăve, however The phase errors caused by the objective lens, W (g), may not be the same for both sets of lattice fringes Any difference would affect the weights of the cos(W (g)) and sin(W (g)) terms in (10.98), so the fringes from the two crystals could be shifted differently To obtain reliable information about the structure of the interface, further analysis of the image is generally required A structural image is a high-resolution image made with several diffracted beams Intersecting sets of fringes are obtained in structural images, producing sets of black or white dots, as in Figs 2.3, 2.23, 2.26 and 2.27 The phase error is generally different for each diffraction used in the image, however, owing to differences in how W (Δk) depends on the order of the diffraction and on the defocus It is not obvious, for example, if columns of atoms should appear as white or black dots, and this appearance can change with the defocus of the objective lens and the thickness of the sample 10.3 Experimental High-Resolution Imaging 545 Weak Phase Object Approximation The physical issues of high-resolution imaging can be understood better by considering a real-space phase function of the sample, qi (x, y) of (10.61), which also includes absorption To understand how the specimen interacts with the phase transfer function of the objective lens, QPTF (Δk) of (10.93), we first take the Fourier transform of (10.61): (10.99) Qi (Δkx , Δky ) = F e−iσ φ(x,y) e−μ(x,y) The weak phase object (WPO) approximation assumes that the specimen is very thin, so σ φ(x, y) and μ(x, y) are very small The exponentials in (10.99) can therefore be linearized: Qi (Δkx , Δky ) = F − i σ φ(x, y) − μ(x, y) (10.100) Likewise we can neglect the small second-order product i σ φ(x, y) μ(x, y), so Qi (Δkx , Δky ) = F [1 − i σ φ(x, y) − μ(x, y)] (10.101) Fourier transforms are distributive, and F [1] = δ(Δkx , Δky ), so Qi (Δkx , Δky ) = δ(Δkx , Δky ) − F [μ(x, y)] − iσF [φ(x, y)] (10.102) This k-space representation of the phase of the electron wavefunction through the sample now can be multiplied conveniently by (10.92), the phase transfer function of the objective lens This gives the “phase transfer modified” electron wavefunction, Qi (Δkx , Δky ): Qi (Δkx , Δky ) = δ(Δkx , Δky ) − F [μ(x, y)] − iσF [φ(x, y)] × eiW(Δkx ,Δky ) (10.103) The important quantity for image formation is of course the intensity The intensity in a real-space image is qi∗ (x, y) qi (x, y) Calculating the complementary k-space intensity function, Itot (Δkx , Δky ), requires a convolution in the Fourier transform representation of the wavefunction and its complex conjugate We calculate Q∗ i (Δkx , Δky ) ∗ Qi (Δkx , Δky ): Itot (Δkx , Δky ) = δ ∗ (Δkx , Δky ) − F ∗ [μ(x, y)] +iσF ∗ [φ(x, y)] e−iW(Δkx ,Δky ) ∗ δ(Δkx , Δky ) − F [μ(x, y)] (10.104) −iσF [φ(x, y)] eiW(Δkx ,Δky ) Equation (10.104) includes nine convolutions Again, however, for thin samples the convolutions, F ∗ [μ(x, y)] ∗ F [μ(x, y)], F ∗ [μ(x, y)] ∗ σF [φ(x, y)], σF ∗ [φ(x, y)] ∗ F [μ(x, y)], and σ F ∗ [φ(x, y)] ∗ F [φ(x, y)], are of second order and are neglected The remaining five convolutions involve delta functions, and can be performed by inspection13 of (10.104): 13 Note that δ(Δkx , Δky ) e−iW(Δkx ,Δky ) = δ(Δkx , Δky ) e−iW(0,0) = δ(Δkx , Δky ) 546 10 High-Resolution TEM Imaging Itot (Δkx , Δky ) = δ(Δkx , Δky ) −F ∗ [μ(x, y)] e−iW(Δkx ,Δky ) − F [μ(x, y)] eiW(Δkx ,Δky ) +iσF ∗ [φ(x, y)] e−iW(Δkx ,Δky ) − iσF [φ(x, y)] eiW(Δkx ,Δky ) (10.105) When the crystal is centro-symmetric, we can set F ∗ [φ(x, y)] = F [φ(x, y)] and F ∗ [μ(x, y)] = F [μ(x, y)], so: Itot (Δkx , Δky ) = δ(Δkx , Δky ) − 2F [μ(x, y)] cos(W (Δkx , Δky )) + 2σF [φ(x, y)] sin(W (Δkx , Δky )) , (10.106) and Itot is real The large first term in (10.106) is the forward beam, peaked at Δk = The second term is the amplitude contrast term, which depends on the absorption of the sample The third term involves the phase shift of the electron wavefront, as provided by the projected potential (cf., (10.68)) Again, as in (10.98) for the two-beam case, the intensity depends on details of the phase error, W (Δkx , Δky ) of (10.92) Real lens characteristics are presented in the next section, but it is often assumed that these characteristics can be loosely approximated as W (Δkx , Δky ) = −π/2, so cos(W (Δkx , Δky )) = and sin(W (Δkx , Δky )) = −1 In many cases absorption is small, also suppressing the second term in (10.106) Equation (10.106) can then be approximated as: Itot (Δkx , Δky ) δ(Δkx , Δky ) − 2σF [φ(x, y)] (10.107) The contrast in the image is therefore approximated as originating from the phase shift of the electron wavefunction through the sample (as caused by (10.68), for example) We say that the specimen is a “weak phase object,” or WPO This approximation is handy for explaining the origin of contrast in a high-resolution image Unfortunately, however, samples are rarely thin enough for this approximation to be valid, and the characteristics of the objective lens cannot be approximated reliably as W (Δkx , Δky ) = −π/2 10.3.3 Lens Characteristics Lens Phase Errors Figure 10.15 shows the phase shift error of a particular microscope, W (Δk) of (10.92), for various values of defocus, Δf The forwardscattered beam at Δk = has a reference phase of for all curves For the smallest values of defocus, there is a rather small error in phase for Δk below about 10 nm−1 , corresponding to a real-space distance of 2π/Δk = 0.6 nm Most samples have many features larger than 0.6 nm For small values of defocus, these larger features show little contrast in the image, since electrons scattered at small Δk are recombined accurately in phase with the forward beam.14 This fact is useful for adjusting the microscope for peak performance 14 So although bright-field and dark-field images have resolution limitations owing to the finite size of the objective aperture, these conventional methods are a good choice for making images of features larger than those on the atomic scale