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High-Resolution Electron Microscopy

Fourth Edition

John C. H. SpenceDepartment of Physics and Astronomy

Arizona State University/LBNL California

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Great Clarendon Street, Oxford, OX2 6DP,United Kingdom

Oxford University Press is a department of the University of Oxford.It furthers the University’s objective of excellence in research, scholarship,

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First Edition published in 1980Second Edition published in 1988Third Edition published in 2003Fourth Edition published in 2013

Previously editions were entitled Experimental high-resolution electron microscopy

Impression: 1

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1

Preliminaries

This chapter is intended to re-orient an electron microscopist familiar with conventionaltransmission electron microscope (TEM) imaging at relatively low resolutions to the special-ized requirements of atomic-resolution electron microscopy. Scanning transmission electronmicroscopy (STEM) is considered in Chapter 8, but much of the following also applies toSTEM.

The conventional TEM bears a close resemblance to an optical microscope in whichimage contrast (intensity variation) is produced by the variation in optical absorption frompoint to point on the specimen. Most electron microscopists interpret their images in asimilar way, taking ‘absorption’ to mean the scattering of electrons outside the objectiveaperture. Unlike X-rays or light, electrons are never actually ‘absorbed’ by a specimen;they can only be lost from the image either by large-angle scattering outside the objectiveaperture or, as a result of energy loss and wavelength change in the specimen, being broughtto a focus on a plane far distant from the viewing screen showing the elastic or zero-lossimage. This out-of-focus ‘inelastic’ or energy-loss image then contributes only a uniformbackground to the in-focus elastic image. Thus image contrast is popularly understood toarise from the creation of a local intensity deficit in regions of large scattering or ‘massthickness’ where these large-angle scattered rays are intercepted by the objective aperture.The theory which describes this process is the theory of incoherent imaging (see, e.g.,Cosslett (1958)).

By comparison, the high-resolution TEM (HRTEM) is a close analogue of the opticalphase-contrast microscope (Mertz 2009). These optical instruments were developed to meetthe needs of biological microscopists who encountered difficulty in obtaining sufficient con-trast from their thinnest specimens. Nineteenth-century microscopists were dismayed to findthe contrast of their biological specimens falling with decreasing specimen thickness. Morethan a century later, improvements in specimen preparation techniques have allowed us tosee exactly the same behaviour in electron microscopy when observing specimens just a fewatomic layers thick. Since the transmission image is necessarily a projection of the speci-men structure in the beam direction (the depth of focus being large—see Section 2.2) theselow-contrast ultra-thin specimens must be used if one wishes to resolve detail at the atomiclevel.

Broadly then, while low-resolution biological microscopy is mainly concerned with theelectron microscope used in a manner analogous to that for a light microscope, this book ismainly devoted to the use of the TEM as a phase-contrast instrument for high-resolutionstudies. For the STEM, we will see that an incoherent mode of atomic-resolution imagingis also possible.

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2 Preliminaries

1.1. Elementary principles of phase-contrast TEM imaging

Fresnel edge fringes, images showing the structure of a thin crystal, and images of smallmolecules are all examples of phase contrast. None of these contrast effects can be explainedusing the ‘mass thickness’ model for imaging with which most microscopists are familiar.All are interference effects. The distinction between interference or phase contrast and con-ventional low-resolution contrast is discussed more fully in Section 6.1. In this section,however, a simple optical bench experiment is described which will give the microscopistsome feel for phase contrast and anticipate the results of many electron-imaging experi-ments. An expensive optical bench is not needed to repeat this experiment—good resultswill be obtained with a large-diameter lens (focal length, f 0, about 14 cm and diameter,d , about 5 cm) and an inexpensive 1.5 mW He–Ne laser. The object-to-lens and lens-to-image distances U and V could then be about 17 and 80 cm, respectively, as indicated inFig. 1.1.

The very thin specimens used in high-resolution electron microscopy can be likened to apiece of glass under an optical microscope. An amorphous carbon film behaves rather like aground-glass screen for optical wavelengths. In a typical high-resolution electron microscopeexperiment this ‘screen’ is used to support a sample of which one wishes to form an image.Figure 1.2 shows the image of a glass microscope slide formed using a single lens and alaser. The laser beam has been expanded (and collimated) using a ×40 optical microscopeobjective lens. The presence of a small molecule or atom is difficult to simulate exactly.However, the glass slide shown contains a small indentation produced by etching the glasswith a small drop of hydrofluoric acid. This indentation will produce a similar contrasteffect to that of an atom imaged by an electron microscope. Figure 1.2 shows a ‘through-focus series’ obtained by moving the lens slightly between exposures. These through-focusseries are commonly taken by high-resolution microscopists to allow the best image to beselected (see Chapter 5). In particular, we notice the lack of contrast in Fig. 1.2(b), theimage recorded at exact Gaussian focus when the object and the image fall on conjugateplanes. Exactly similar effects are seen on the through-focus series of electron micrographs

P

L1

L2 L3 M FOU VYf0

f0

Figure 1.1 The optical bench arrangement used to record the images shown in Figs. 1.2–1.4. Here

L1 is a ×40 microscope objective lens at the focus of which is placed a pin-hole aperture P. Lenses

L2 and L3 have a focal length of f 0 = 14 cm. The object is shown at 0 and the film plane at F;

distances are Y = 30 cm, U = 17 cm, and V = 80 cm. The pin-hole aperture is used as a spatial

filter to provide more uniform illumination. Back-focal plane masks may be inserted at M.

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Elementary principles of TEM imaging 3

(a)

(b)

(c)

Figure 1.2 Optical through-focus series showing the effect of focus changes on the image of a small

indentation in a glass plate (phase object). The image at (a) was recorded under-focus, that is,

with the object too close to the lens L3. It shows a bright fringe surrounding the indentation

similar to that seen on electron micrographs of small holes; image (b) is recorded at exact focus

and shows only very faint contrast; image (c) is recorded at an over-focus setting (object too far

from L3) and so shows a dark Fresnel fringe outlining the indentation. The background fringes

arise in the illuminating system. Compare these with the electron micrographs in Fig. 6.7.

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4 Preliminaries

Figure 1.3 An image recorded under identical conditions to that shown in Fig. 1.2(a), with the

laser source replaced with a tungsten lamp focused onto the object (critical illumination). The

faint contrast seen is due to the preservation of some coherence in the illumination introduced by

limiting the size of lens L2. This contrast disappears completely if a large lens is used. Variations

in the size of this lens (or an aperture near it) are analogous to changes in the size of the second

condenser aperture in an electron microscope.

of single atoms shown in Fig. 6.7 . Figure 1.3 shows the same object imaged using a conven-tional tungsten lamp-bulb as the source of illumination to provide ‘incoherent’ illuminationconditions. Despite wide changes in focus, little contrast appears.

If this analogy between the indented glass slide and a high-resolution specimen inelectron microscopy holds, this simple experiment suggests that our high-resolution spe-cimens will be imaged with strong contrast only if a coherent source of illumination isused and if images are recorded slightly out of focus. These conclusions are borne out inpractice, and the importance of high-intensity coherent electron sources for high-resolutionmicroscopy is emphasized throughout this book. Two chapters have been devoted to thetopic—Chapter 12 deals with experimental aspects and Chapter 4 describes the elementarycoherence theory.

To return to our optical analogue. Given coherent illumination the question arises as towhether methods other than the introduction of a focusing error can be used to produceimage contrast. During the last century many empirical methods were indeed developed toenhance the contrast of images such as that in Fig. 1.2(b). These included: (1) reducingthe size of the objective aperture as in Fig. 1.4(a); (2) introducing a focusing error as inFig. 1.2(a); (3) simple interventions in the lens back-focal plane as in Fig. 1.4(b), whereSchlieren contrast is shown (the back-focal plane is approximately the plane of the objectiveaperture for an electron microscope); and (4) the use of back-focal plane phase plates, similarto the Zernike phase plate used in optical microscopy.

All of these methods have their parallel in electron microscopy. The first is not useful,since image resolution is necessarily limited. The second is the usual method of high-resolution electron microscopy and is discussed in more detail in Chapters 3 and 12. Methodsbased on interventions in the lens back-focal plane have appeared from time to time (see,

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Elementary principles of TEM imaging 5

(a)

(b)

Figure 1.4 Using a laser source, both these images were recorded at exact focus (exactly as in Fig.

1.2(b)). Yet in both, the outline of the indentation can be seen. In (a), a small aperture has been

placed on the axis at M, severely limiting the image resolution. On removing this aperture the

image contrast disappears. The use of a small aperture at M (the back-focal plane) is analogous

to the normal low-resolution method of obtaining contrast in biological electron microscopy. In (b)

a razor blade has been placed across the beam at M, thus preventing exactly half the diffraction

pattern from contributing to the image. The resulting image is approximately proportional to the

derivative of the phase shift introduced by the object taken in a direction normal to the edge of

the razor blade. Notice the fine fringes inside the edge of the indentation arising from multiple

reflection within the glass slide.

e.g., Spence (1974)), while there have been steady improvements in the construction ofZernike phase-plates over the years for electron microscopes (corresponding to the use ofback-focal plane phase plates), as reviewed by Hall et al . (2011). Carbon films or appliedelectric fields across the back-focal plane have been used—in one scheme, for example, aninfrared laser beam, focused onto the central spot of the diffraction pattern, will cause therequired 90◦ phase shift in the central undiffracted electron beam (Muller et al . 2010).

In both electron and optical microscopy the reasons for the lack of contrast at exactfocus are the same—these thin specimens (‘phase objects’) affect only the phase of the wavetransmitted by the specimen and not its amplitude. That is, they behave like a mediumof variable refractive index (see Section 2.1). It is this variation in refractive index frompoint to point across the specimen (proportional to the specimen’s atomic potential in volts

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6 Preliminaries

for electron microscopy) which must be converted into intensity variations in the image ifwe are to ‘see’, for example, atoms in the electron microscope. Through the reciprocitytheorem, discussed in Section 8.1, all these considerations apply to the imaging of singleatoms in the STEM in bright-field mode.

We can sharpen these ideas by comparing the simplest mathematical expressions forimaging under coherent and incoherent illumination. For the piece of glass shown in Fig. 1.2,the phase of the wave transmitted through the glass differs from that of an unobstructedreference wave by 2π(n − 1)/λ times the thickness of the glass, where n is the refractiveindex of the glass. If t(x ) is the thickness of the glass at the object point x , the amplitudeof the optical wave leaving the glass is given from elementary optics (see, e.g., Lipson andLipson (1969)) by

f(x) = exp(−2πint(x)/λ) (1.1)

The corresponding expression for electrons is derived in Section 2.1. Optical imaging theory(Goodman 2004) gives the image intensity, as

I(x)c = |f(x) ∗ S(x)|2 (1.2)

for coherent illumination and

I(x)i = |f(x)|2 ∗ |S(x)|2 (1.3)

for incoherent illumination. In these equations the function S (x ) is known as the instru-mental impulse response, and gives the image amplitude which would be observed if onewere to form an image of a point object much smaller than the resolution limit of themicroscope. Such an object does not exist for electron microscopy; however, the expectedform of S (x ) for one modern electron microscope can be seen in Fig. 3.8. This functionspecifies all the instrumental imperfections and parameters such as objective aperture size(which determines the diffraction limit), the lens aberrations, and the magnitude of anyfocusing error. The asterisk in eqns (1.2) and (1.3) indicates the mathematical operationof convolution (see Section 3.1), which results in a smearing or broadening of the objectfunction f (x ). It is not possible to resolve detail finer than the width of the function S (x ).Using eqn (1.1) in eqns (1.2) and (1.3) we see that under incoherent illumination the imageintensity from such a phase object does not vary with position in the object, since

| exp{−2πit(x)n/λ} |2 = 1 (1.4)

Only by using coherent illumination and an ‘imperfect’ microscope can we hope to obtaincontrast variations in the image of a specimen showing only variations in refractive index.In high-resolution electron microscopy of thin specimens the accurate control of illuminationcoherence and defect of focus are crucial for success. The amount of fine detail in a high-resolution TEM micrograph increases dramatically with improved coherence of illumination,while completely misleading detail may be observed in images recorded at the wrong focussetting.

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Elementary principles of TEM imaging 7

1 nm

(a) (b)

Figure 1.5 Two electron microscope images of amorphous carbon films recorded at the same focus

setting but using different condenser apertures. In (a) a small second condenser aperture has been

used, resulting in an image showing high contrast and fine detail. This contrast is lost in (b), where

a large aperture has been used. This effect is seen even more clearly in optical diffractograms (see

Chapter 4).

To demonstrate the effect of coherence on TEM image quality at high resolution, Fig. 1.5shows images of an amorphous carbon film using widely different illumination coherenceconditions but otherwise identical conditions. The loss of fine detail in the ‘incoherent’image is clear. In practice, for a microscope fitted with a conventional hair-pin filament, theillumination coherence is determined by the size of the second condenser lens aperture, asmall aperture producing high coherence. With a sufficiently small electron source (such as afield-emission source) the illumination may be almost perfectly coherent and so independentof the choice of second condenser (illuminating) aperture.

In most cases of practical interest the imaging is partially coherent. By this we looselymean that for object detail below a certain size X c we can use the model of coherentphase contrast imaging (see Fig. 1.2) while for detail much larger than X c the imaging isincoherent. The distance X c is given approximately by the electron wavelength divided bythe semi-angle subtended by the second condenser aperture at the specimen, when using ahair-pin filament. In Section 10.7 a fuller discussion of methods for measuring X c is givenwhich includes the effect of the objective lens pre-field. Since the width of Fresnel edgefringes commonly seen on micrographs is approximately equal to X c, we can adopt therough rule of thumb that the more complicated through-focus effects of phase contrastdescribed in Section 6.2 will become important whenever one is interested in image detailmuch finer than the width of these fringes.

To summarize, the main changes in emphasis for microscopists accustomed to conven-tional TEM imaging and considering a high-resolution project are as follows: in conventional

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8 Preliminaries

imaging we deal with ‘thick’, possibly stained, specimens using a small objective apertureand a large condenser aperture. Focusing presents no special problems. In high-resolutionphase-contrast imaging we deal with the thinnest possible specimens (unstained), using alarger objective aperture of optimum dimensions and a small illuminating aperture. Thechoice of focus becomes crucial and must be guided either by experience with computedimages or by the presence of a small region of specimen of known structure in the neigh-bourhood of the wanted structure. This image simulation experience indicates that for thevast majority of large-unit-cell crystals imaged at a resolution no higher than 0.20 nmthe choice of focus which gives a readily interpretable image containing no false detail isbetween 20 and 60 nm under-focus (weakened lens) for microscopes with a spherical aber-ration constant between 1 and 2 mm. A calibrated focus control is therefore needed to setthis focus condition. Methods for measuring the focus increments are given in Section 10.1.For aberration-corrected instruments, the choice of aberration-free conditions produces anin-focus image with zero contrast, as we see from eqn (1.4). Section 7.4 describes theoptimum choice of defocus and aberration constants for best resolution and contrast forthese instruments. The limits on specimen thickness are discussed in Chapter 5.

1.2. Instrumental requirements for high resolution

The major manufacturers of electron microscopes offer instruments that are closely matchedin performance. All are capable of giving point resolution better than 0.2 nm for bright-fieldimages. The use of side-entry eucentric goniometer stages, of the kind used by biologists formany years, for high-resolution work greatly facilitates the procedure of orienting a smallcrystal, since the lateral movement accompanying tilting of the specimen is minimized bythese stages. However, the drift rate due to thermal effects using these stages has beenconsiderably higher than that for top-entry stages.

A laboratory which has recently purchased a TEM and wishes to use it for high-resolution studies should consider the following points. This list is intended as a roughchecklist to refer the reader to further discussion of these topics in other chapters.

1. The microscope site must be acceptable. Mechanical vibration, stray magnetic fields,and room temperature must all be within acceptable limits. These factors are discussed inChapter 11.

2. A reliable supply of clean cooling water at constant temperature and pressure must beassured (see Chapter 11). In installations where internal recirculating water is not availableand the external supply is impure it is common to find both the specimen drift rate andillumination stability deteriorating over a period of weeks as the water filter clogs up,causing the pressure of the cooling water to fall and leading to fluctuations in temperatureof the specimen stage and lens. A separate closed-circuit water supply and external heatexchanger is the best solution for high-resolution work, with the water supplied only to themicroscope. Vibration from the heat exchanger pumps must be considered; these should bein a separate room from the microscope. Despite the use of distilled water, algae are boundto form in the water supply. Some manufacturers warn against the use of algae inhibitorsas these may corrode the lens-cooling jacket, repairs to which are very expensive. About

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Instrumental requirements 9

all that can be done is to replace the water periodically. Thermal stage drift is one of themost serious problems for high-resolution imaging, particularly for long-exposure dark-fieldimages. If external hard water is used it is essential that the condition of the water filter ischecked regularly and replaced if dirty . If the microscope’s inbuilt thermoregulator is usedit is equally important to check every few days that it is operating within the temperaturelimits for which it is designed.

3. In order to record an image at a specified focus defect it will be necessary to measurethe change in focus between focus control steps (‘clicks’) using the methods of Section 10.1.In addition, the spherical aberration constant C s must be known for the optimum objectivelens excitation. This should be less than 2 mm at 100 kV if high-resolution results areexpected. Methods for measuring C s are given in Section 10.2. Microscope calibration—the measurement of defocus increments, spherical aberration constant, and objective lenscurrent—is therefore an important first step. Once the optimum specimen position in thelens gap has been found, the objective lens current needed to focus the specimen at thisposition should be noted, and the highest-quality images recorded near this lens current.This is not always possible when using a tilting specimen holder if a non-axial object point isrequired, since the specimen height (and hence the objective lens current needed for focus)varies across the inclined specimen.

4. The resolution obtained in a transmission image depends, amongst other things, onthe specimen position in the objective lens. The optimum ‘specimen height’ must be foundby trial and error. In reducing the specimen height and so increasing the objective lens cur-rent needed for focus, the lens focal length and spherical aberration constant are reduced,leading to improved resolution. Figure 2.12(a) shows the dependence of spherical aberrationon lens excitation. A second effect, however, results from the depth of the objective lenspre-field, which increases as the specimen is immersed more deeply into the lens gap. Thisincreases the overall demagnification of the illumination system and increases its angularmagnification (see Section 2.2). The result is an increase in the size of the diffraction spots ofa crystalline specimen which, for a fixed illumination aperture, allows the unwanted effectof spherical aberration to act over a larger range of angles (see Section 3.3). Finally, asshown in Fig. 2.12(a), for many lens designs the chromatic aberration coefficient C c passesthrough a minimum as a function of lens excitation, and this parameter affects both thecontrast and resolution of fine image detail. The complicated interaction between all thesefactors which depend on specimen position can best be understood using the ‘dampingenvelope’ concept described in Section 4.2 and Appendix 3. This ‘damping envelope’ con-trols the information resolution limit (loosely referred to by manufacturers as the ‘line’ or‘lattice’ resolution) of the instrument and depends chiefly on the size of the illuminationaperture and C c. The point resolution, however, is determined by spherical aberration.A method has been described which would allow both these important resolution limitsto be measured as a function of specimen position in the lens bore through an analysis ofoptical diffractogram pairs (see Section 10.6). A systematic analysis of this kind, however,represents a sizeable research project, involving many practical difficulties. For example,in such an analysis, diffractograms are required at similar focus defects for a range ofspecimen positions, yet the ‘reference focus’ used to establish a known focus defect (seeSection 12.5) itself depends on C s, which varies with specimen position. In addition, boththe focal increment corresponding to a single step on the focus control and the chromatic

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10 Preliminaries

aberration constant (which affects the ‘size’ of diffractogram ring patterns) depend on thespecimen position. For preliminary work, then, a simpler procedure is to record images ofa crystal with a large unit cell at the Scherzer focus for a range of specimen heights and toselect the highest-quality image, judging this by eye. Experience in comparing computedand experimental images of crystals with large unit cells shows that a useful judgementof image quality (a combination of contrast, point, and line resolution) can be made byeye. Once this near-optimum specimen position has been found, further refinements can bemade using the computer image-matching technique together with diffractogram analysis.To obtain the required range of specimen heights on top-entry machines it is necessary tofit small brass washers above or below the specimen. A shallow threaded cap to fit over thespecimen may also be needed to bring the tilted specimen closer to the lower cold-finger.The measured objective lens current needed to bring the image into focus is then used as ameasure of specimen height, and must be recorded for each image. The specimen height andcorresponding lens current needed to give images of the highest quality can then be determ-ined and permanently recorded for the particular electron microscope. Crystals with largeunit cells have the useful property that the highest-contrast images are usually producednear the Scherzer focus, so that this focus setting can be found routinely by experiencedmicroscopists working with these materials.

5. A vacuum of 0.5 × 10−7 Torr or better is needed (measured in the rear pumping line).The simplest way to trace vacuum leaks is to use a partial pressure gauge (see Section 11.5).A microscope which is to be used for high-resolution imaging should be fitted permanentlywith such a gauge, while plasma-cleaning apparatus for the sample holder is also crucial.Inexpensive gauges of the radiofrequency quadrupole type are useful. In a clean, well-out-gassed microscope with no serious leaks and a vacuum of 0.5 × 10−7 Torr the major sourceof contamination is frequently the specimen itself. Always compare the contamination ratesfor several different specimens before concluding that the microscope itself is causing acontamination problem.

6. The high-voltage supply of the microscope must be sufficiently stable to allow high-resolution images to be obtained. Given a sufficiently stable high-voltage supply (seeSections 11.2 and 2.8.2) this generally requires a scrupulously clean gun chamber andWehnelt cylinder if a field-emission gun is not used. Methods for observing high-voltageinstability are discussed in Section 12.2.

7. The room containing the microscope must be easily darkened completely, and aroom-light dimmer control needs to be fitted within arm’s reach of the operator’s chair.Specimen changes can then be made in dim lighting so that the user does not lose the slowchemical dark adaptation of the eyes. The newest generation of atomic-resolution machinesare controlled digitally from a neighboring room.

1.3. First experiments

This section outlines a procedure which will enable a microscopist new to high-resolutionwork to become familiar with the techniques of high-resolution TEM and to practise therequired skills of focusing, astigmatism correction, and image examination. The test speci-men used consists of small clusters of heavy atoms supported on a thin carbon film. Themicroscope is assumed to satisfy the conditions of Section 1.2 and the focal increments andspherical aberration constant are presumed known.

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References 11

1. Specimens, consisting of small clusters of heavy atoms lying on a thin carbon substratefilm, can be purchased from suppliers.

2. Once a suitable specimen area has been found, bright-field TEM images can berecorded at high magnification (400 000 or more) using a charge-coupled device (CCD)camera. Chapter 12 gives details of all necessary precautions. A microscope which has beenswitched on in the morning, with all cold-traps kept filled with liquid nitrogen, should be inthermal equilibrium by early afternoon when serious work can commence. The morning canbe profitably spent examining possible specimens. The cold-traps must not be allowed toboil dry during a lunch break. The thermal stability and cleanliness of the three componentsin the objective lens pole-piece gap (objective aperture, cold-finger, and specimen) are of theutmost importance in high-resolution electron microscopy . The best arrangement is to leavethe microscope running through the week, fitted with an automatic liquid-nitrogen pumpfor the cold-traps. Record images without an objective aperture inserted (so that resolutionis limited by incoherent instabilities—see Section 4.2) and also using an aperture whosesize is given by eqn (6.16b). The semi-angle θap subtended by the objective aperture can bemeasured by taking a double exposure of the diffraction pattern of some continuous gold filmwith, and without, the aperture in place (see Section 10.2). A check of the following mustbe made before taking pictures: (1) condenser astigmatism (for maximum image intensity);(2) gun tilt; (3) current or voltage centre (see Section 12.2); (4) high-voltage stability(see Section 11.2); (5) absence of thermal contact between specimen and cold-finger (seeSection 12.2); (6) cleanliness of objective aperture; (7) specimen drift (see Section 11.4);and (8) contamination (see Section 11.5).

Use the minimum-contrast condition (see Section 12.5) to correct astigmatism and takeseveral bright-field images in the neighbourhood of the focus value given by eqn (6.16a).A through-focus series about the minimum contrast focus in steps of, say, 20 nm will showthe characteristic change from a dark over-focus Fresnel fringe around an atom cluster toa bright fringe in the under-focus images.

3. Digital diffractograms of the images should be obtained as described in Section 10.6.The measured diameter of the rings seen in these can be used with the simple computerprogram given in Appendix 1 to find the focus setting for each micrograph and themicroscope’s spherical aberration constant. Alternatively, the images may be analysedusing commercial software. These diffractograms reveal at a glance the presence ofastigmatism or specimen movement during the exposure (drift—see Section 10.6). Thisimmediate ‘feedback’ is essential for a microscopist learning the skills of astigmatismcorrection and focusing. With practice the microscopist will become adept at findingthe minimum-contrast condition, correcting astigmatism, and resetting the focus controla fixed number of ‘clicks’ toward the under-focus side to obtain images of the highestcontrast and resolution using these test samples.

References

Cosslett, V. E. (1958). Quantitative aspects of electron staining. J. Roy. Microsc. Soc. 78, 18.Goodman, J. W. (2004). Introduction to Fourier optics. Third Edition. McGraw-Hill, New York.Hall, R. J., Nogales, E., and Glaeser, R. M. (2011). Accurate modeling of single-particle cryo-

EM images quantifies the benefits expected from using Zernike phase contrast. J. Struct. Biol .174, 468.

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12 Preliminaries

Lipson, S. G. and Lipson, H. (1969). Optical physics. Cambridge University Press, London.Mertz, J. (2009). Introduction to optical microscopy . Roberts and Co., Greenwood Village, CO.Muller, H., Danev, J., Spence, J., Padmore, H., and Glaeser, R. M. (2010). Design of an electron

microscope phase plate using a focussed continuous-wave laser. New J. Phys. 12, 073011.Spence, J. C. H. (1974). Complex image determination in the electron microscope. Opt. Acta

21, 835.