I. Limits to Resolution

A. Abbe--Fourier Optics -- apply diffraction theory to coherent imaging. The lens focuses the diffracted rays from the object to form the Fraunhofer diffraction pattern of the object at the back focal plane of the lens (Fig. 1a); this is a spatial frequency analysis of the object, i.e. a Fourier Transform. The image is formed by a 2nd Fourier transformation and is a magnified inverted image of the object. The magnification is R1/Ro, the ratio of the lens-image distance the lens-object distance. The image is not a "true" representation of the object because:

1. lens aberrations alter the Fourier Transform

2. the lens aperture (expressed by its half angle, a) may limit how many diffraction orders pass
===> resolution is limited

Fig. 1 a b

Quantitation: If 2 image points are separated by a distance, d, then their first diffraction order will occur at an angle, q which will permit the path difference for light waves diffracted from the two points to constructively interfere; this means that the path difference, d, for the two waves must equal the wavelength, l, of the radiation or d = d n sin(q) = l where n is the index of refraction of the medium (Fig. 1b). The resolution limit is (approximately) equal to the d which corresponds to a first order diffraction at q = a where a is the aperture half angle of the objective lens. Substituting a for q in the equation above gives:

n sin(a) is called the Numerical Aperture or N.A. To be quantitatively correct, l must be multiplied by a factor of 0.5 - l which depends upon the coherency of illumination, the type of contrast generated (phase, fluorescence, interference etc.). Thus, the resolution depends upon both the wavelength of radiation used to image the object and upon the aperture of the imaging system.

B. Light Microscopy

1. Wavelength, l = 4000 Å

2. Numerical Aperature = 1.6--Resolution, d = 2500 Å

Further increases in N.A. are not possible, therefore achieving higher (finer) resolution requires smaller l. UV radiation isn't much improvement and we don't know how to make lenses to focus X-rays and g-rays.

C. Electron Microscopy

1. deBroglie discovered that moving particles have a wave nature with l = h / mv where h = Planck's const., m = particle mass, and v = particle velocity. Electrons have a charge and can be accelerated in an electric potential field as well as focused by electric or magnetic fields. An electron accelerated in a potential of V volts has kinetic energy 1/2m v2 = e V where e is the charge on the electron. Solving for v and substituting into deBroglie's equation (and expanding in V to account for the fact that the electron mass is different when moving than when at rest):

Thus, for V = 100,000 V (a typical value) l = 0.037 Å

for V = 1,000,000 V (High Voltage EM) l = 0.0087 Å

2. What resolution can be expected?

  1. electron lenses are poor and are therefore designed with very small apertures, e.g. a = 0.02 radians
  2. d = 2 - 4 Å ==> Theoretical resolution is still quite small and is achieved with electron microscopes.
II. Electron Microscope Design - All optics at < 10-5 torr.

A. Source--Electron Gun

1. Variable Bias--hot tungsten hairpin filament--most common type (Fig. 2a). The filament is heated by a current, IF, and electrons are drawn off by the potential difference between the filament and the anode. The filament is surrounded by a shield or gun cap (also called a Wehnelt) which is kept several hundred volts more negative than the filament by a variable bias resistor which connects the two. This potential difference and the design of the Wehnelt focuses the electrons at a crossover point between the front of the Wehnelt and the anode, this point is the effective source of illumination for the system. Brightness of the source increases as IF is increased up to a Saturation point when further current increases don't increase brightness but does shorten filament life (Fig. 2b).

2. Pointed Filament -- Brightness and coherence increased by decreasing size of the area from which electrons are drawn. Pointed filaments may be made from a hairpin filament with a small tungsten crystal or etched piece of tungsten wire welded onto it. A better solution is a pointed filament made of LaB6; such a filament can be 100x brighter (usually more like 10x) than a normal hairpin filament..

Fig. 2 a b

3. Field Emission Gun (cold cathode) -- A very fine tip which is usually not heated. Instead, electrons are drawn from the tip by a sharp potential drop (electrons "tunnel" out). This type of source is very difficult to construct and maintain partially because it requires ultra-high vacuum technology, but it is 104 x brighter than a conventional filament and produces a beam of electrons with a very small spread of energies (very monochromatic and very coherent).

B. Lenses

1. Electrostatic -- electrons are focused by an electric field. The variable bias electron gun is the only electrostatic lens used in an electron microscope

2. Magnetic -- All other lenses used are electro-magnetic. The magnetic field produced by current flowing through the Cu wire lens coil is concentrated by soft iron pole pieces which direct the lines of force (Fig. 3a). Electrons entering the lens off axis are focused along a spiral path to a point. As with light microscopes, the objective lens is the most critical; typically, the objective lens has a bore of approximately 2 mm and a focal length of 1-2 mm. Magnetic lenses cannot be made or maintained with completely radially symmetric fields. Therefore, coils are included to correct the asymmetry or the "astigmatism" of the lens ==> Stigmator coils--these are adjusted so that the lens has the same focal length in all directions.

Fig. 3 a b

C. Layout --Conventional Transmission Electron Microscope (CTEM) everything must be in a vacuum of at least 10-4 - 10-5 torr (preferably much lower, e.g. 10-7-10-8torr) to prevent electron scattering by gas molecules.

1. Illumination--electron gun plus 2 condenser lenses (and 2 condenser lens apertures). The condenser lens system must:

  1. focus a reduced image of the gun crossover on the specimen to make a small bright probe or
  2. provide nearly parallel illumination at low intensity with high coherency.
2. Objective Lens--the imaging Lens (most critical) -- for Philips EM 420 Super Twin Objective Lens
f = 1.7 mm Cs = 1.2 mm (Coeff. of Spherical aberration) Cc = 1.2 mm (Coeff. of chromatic aberration) ===> resolution = 3.0 Å point-to-point and 1.4 Å line

3. Diffraction Lens--magnify 1st stage image from the objective. lens. The diffraction lens can also be adjusted to image the objective lens back focal plane; this is the electron diffraction pattern which may then be magnified by other lenses.

4. Intermediate plus 1-2 Projection Lenses--these provide further magnification (like the eyepiece of a light microscope).

5. Image on Fluorescent Screen or Film. Film is usually below the fluorescent screen (some microscopes have 35 mm film above the screen in the column). The fluorescent screen is used for viewing, selecting areas to photograph, focusing, and astigmatism correction. It is then flipped up so that the shutter can expose the film below.

6. Magnification by objective lens is fixed since the specimen is always (usually) in the same position. Mag of the projector lens is fixed because the image plane is always on the film or fluorescent screen. Variation is accomplished with diffraction and intermediate lenses; typically the magnification can be varied from 100x - 500000x.

D. Scanning Transmission Electron Microscope (STEM)

1. Source -- field emission gun to provide maximal brightness + coherence.

2. Objective lens forms small electron probe (2 - 5 Å diameter) which is scanned rapidly in a 2-dimensional raster pattern across the specimen by a set of scanning coils. Most electrons pass through the specimen and are undeflected, they are confined to a relatively narrow solid angle determined by the angle of incidence of the small scanning probe. Some electrons are inelastically scattered; most of these are scattered at relatively low angles in the same region as unscattered electrons. Elastically scattered electrons are scattered over a large solid angle.

3. Detectors below--The signal from elastically scattered electrons can be collected by an annular detector below the specimen. A central detector is used to collect the signal from unscattered and from inelastically scattered electrons. Inelastic and unscattered electrons can be separated according to their different energies (wavelengths) by placing a magnetic prism above the central detector. As the probe is scanned across the specimen, the analog signal from the detector(s), usually the annular detector, is displayed synchronously on a cathode ray tube (CRT). The magnification is = (CRT raster)/(electron probe raster). The analog signal can be converted to digital form and stored on computer disk or magnetic tape. The signal from the annular detector is proportional to the number of elastically scattered electrons which is proportional to the mass of the specimen at the probe position ==> STEM is a very sensitive microbalance which can be used to measure the mass of single molecules!

E. Scanning Electron Microscope -- SEM (Fig. 5)

1. Similar to STEM except used with thick specimens --> images surfaces. Probe is scanned across the specimen. This results in stimulation of emission of secondary electrons which have much lower energy; these are attracted to the secondary electron detector by an electrostatic field, and the signal from the detector is displayed on a CRT. High energy electrons from the scanned probe may also be scattered back in the direction from which they came and be detected by a backscattered electron detector placed up near the final lens. The signal from the backscattered electron detector can also be displayed on the CRT, and the resulting images contains information on atomic number of the different areas of the specimen.

2. Generally a lower resolution image (50 - 100 Å) -- limited by probe size and (more significantly) by area from which secondary and backscattered electrons are emitted.

III. Contrast

A. Interactions of Electrons with Specimen

1. Elastic Scattering--negligible loss of energy ==> don't damage and are scattered over relatively large angles.

2. Inelastic Scattering--electrons lose energy (10's of eV's) which is transferred to electrons of the specimen forming ions and free radicals breaking chemical bonds. This is the source of radiation damage which is the principal limitation to high resolution structure determination. Since inelastically scattered electrons have lost energy, their wavelength is altered ==> chromatic aberration which is limits resolution with very thick specimens. The amount of energy lost depends upon the atom with which the electron interacts ==> information on elemental composition. Electrons are scattered inelastically over relatively narrow angles.

B. Bright Field CTEM -- How does the optical system modify the Fourier transform of the object in the back focal plane of the objective lens to generate contrast (see Fig.6).

1. Amplitude Contrast--some elastically scattered electrons fall outside the objective lens aperture and are missing from the image. This is similar to absorption contrast in light microscopy although virtually no electrons are actually absorbed by the thin specimens used in transmission electron microscopy. Relatively low resolution contrast.

2 Phase Contrast--a thin specimen (a few 100's of Å thick) acts as an almost pure phase object (see Fig. 6); this is analogous to phase contrast in light microscopy.

  1. phases of elastically scattered electrons are shifted as they pass through the specimen; the amount of phase shift is proportional to the projection of the electrostatic potential field (created by the electron density of the specimen atoms) of the specimen along the z axis (the optic axis). In order to image the phase object, we want to shift the phases of the diffracted electrons by a quarter wave, p/2 radians, so they will be 180° out of phase and can interfere destructively with unscattered electrons. In light microscopy this would be accomplished with an annular phase plate in the diffraction plane of the objective lens; this is not very practical in electron microscopy. However...
  2. phases of diffracted electrons in the back focal plane of the objective lens (the diffraction plane) are also shifted by:
phase shift as a function of the diffraction angle, a: 

c(a) is radially symmetric if astigmatism is corrected (i.e. Df is the same in all directions). Note that the coefficient of spherical aberration, Cs, changes the phase in a negative direction with a4 dependence while ç changes phase in a positive direction with an a2 dependence. Since Cs is fixed, can we choose a Df which gives a p/2 phase shift over a large range of a (diffraction angle) and balance one effect with the other? The answer is yes, approximately. For a resolution down to 3 - 4 Å, a Df of 1000 Å (100 nm) produces the appropriate phase shift; at finer resolution (< 3 Å) the value of c(a) oscillates rapidly, but the effects of this can be removed by employing an objective aperture of the proper size (see Fig. 6B and note that by convention sin[c(a)] = -1 for proper phase contrast). This Df provides little phase contrast at medium resolution (ca 20 - 30 Å) important in studies of biological specimens. A better choice of Df in biological electron microscopy is approximately 5000 Å (Fig. 6B) which gives more rapid oscillations of c(a). Amplitude contrast from electrons blocked by the objective aperture contributes at lower resolution (> 50 Å).

C. Darkfield


  1. central beam stop in the objective aperture -- such apertures are hard to make and position and also suffer from the problem of charge building up on the central stop (Fig.7a)
  2. tilted beam -- this is the most common technique. The incident beam is tilted so that it falls on the outer edge of a normal objective aperture. Many scattered electron pass through the objective aperture and are used to form an image (Fig. 7b)
  3. hollow cone illumination -- a variation of the tilted beam technique in which the incident beam is a hollow cone converging to a point at the specimen plane. The undiffracted electrons form a second hollow cone below the specimen which is intercepted by a normal objective aperture.
  4. All methods suffer to differing extents from the fact that a large portion of scattered electrons are also stopped by the objective aperture so that much potential information is lost. This leads to a requirement for long exposures which results in much greater beam damage.

2. STEM--signal from annular detector (elastically scattered electrons) displayed on CRT synchronized with STEM scan (Fig. 4). This is certainly the most efficient geometry for darkfield electron microscopy since the annular detector can be designed to collect virtually all of the elastically scattered electrons and can also exclude almost all inelastically scattered electrons which degrade CTEM images by increasing chromatic aberration. STEM darkfield and CTEM brightfield are roughly equivalent in efficiency of data collection.

D. Detection

1. Photographic Emulsion--most efficient

  1. linear relationship between optical density and electron exposure up to approximately 2.5 OD
  2. with the most sensitive films, each electron can produce one or more grains on the film
  3. low cost -- 40-cent sheet of film contains 107 - 108 pixels ==> 10-6 cents/pixel (each pixel contains at least 1 byte of information ==> 1 film = 10 - 100 Mbytes!).
  4. contaminates microscope column with H2O
  5. requires expensive film digitizer for digital image processing
2. Image Intensifier/TV (TV image of high resolution fluorescent screen) -- now readily available but not widely used. Allows online signal processing, but data storage is more expensive (magnetic disks or tape). Optical disks may be the solution.

3. Charge Coupled Device (CCD)--2-dimensional array of solid state detectors each of which builds up charge when light strikes them; this is coupled to electron image by something which converts electrons to photons. (CCD's are commonly used in video cameras; very high quality and sensitive CCD's are used in light microscopy and astronomical telescopes). Slow-scan cooled CCD arrays are very sensitive detectors with very large dynamic range; their resolution is limited, however, by the CCD array size

E. Depth of Field

1. Small objective lens aperture---depth of field > one micron

2. All of the specimen is in focus -- no optical sectioning is possible

  1. image is a 2-dimensional projection of a 3D object ==> confusing overlap of details along z-axis
  2. sorting out overlap requires 3-dimensional reconstruction
IV. Analytical Electron Microscopy--Elemental Composition

A. Inelastic Interactions--provide compositional information by excitation of various energy states of the specimen (Fig. 8). Process involves ionization of atoms.

Fig. 8 a b

1. Low energy loss (< 50 eV)--least useful in analysis; could, in principal, provide details of molecular structure, but hasn't proven practical

  1. excitation of valence and conduction electrons or vibrational states; electrons in molecular orbitals.
  2. great majority of inelastically scattered electrons are of this type
  3. mean free path 500-1000 Å for this type of interaction ==> on average an electron will travel 500-1000Å before being inelastically scattered with low energy loss ==> approximately half of incident electrons will be inelastically scattered when passing through a 1000 Å thick specimen.
2. High energy loss (50 - 1000 eV)--most useful in analytical EM
  1. inner shell ionization edges--normally K shell but also L and M shells. Incident high energy electron knocks inner shell electron out of its orbital. The amount of energy lost by the incident electron is characteristic of the energy of the inner shell electron, that is, characteristic of the element being ionized.
  2. mean free path = several microns --occur much less frequently so very high exposures must be used resulting in radiation damage to the specimen.

Fig. 9 a b

B. Electron Energy Loss Spectroscopy--EELS -- study the primary event

1. Disperse electron according to energy--magnetic spectrometer (Fig. 9a).

  1. efficiency of detection = 20-50%
  2. resolution--typically 10-20 eV---can be 1-2 eV
2. Minimum detectable mass--- 10-20-10-21 gm = 500 - 50 carbon atoms.

3. Study absorption edge fine structure--information on local atomic environment

4. Data Collection:

  1. Serial -- most common mode; scan through the energy spectrum ==> inefficient
  2. Parallel -- new detectors can record all energies at once
C. X-Ray Fluorescence--Energy Dispersive X-Ray Spectroscopy (EDS)

1. An inner shell electron is ejected (e.g. K shell)--This is the primary event which leads to a characteristic loss of energy by the incident electron that is the basis for EELS (see above).

  1. This electron is replaced by an electron from higher energy shell (e.g. L)

  2. An X-ray photon emitted; the energy (i.e. wavelength) of the emitted photon is equal to the difference in energy of the orbitals and (as in the case of the energy lost by the incident electron, e.g. EELS) is characteristic of the element undergoing ionization. Fig. 8b. Energy of emitted X-ray will be less than energy lost by incident electron.
2. Detection -- X-rays are emitted over all possible angles ==> detectors must be placed as close as possible in order to collect the maximum number of emitted X-rays.
  1. Wavelength Dispersive Spectrometer -- Disperse X-rays using a crystal monochromater. As in a UV/Vis spectrophotometer, one can select a single wavelength to measure by positioning a slit which only lets a narrow band of wavelengths pass. Provides very high resolution (ca. 10 eV) but is very inefficient as one only measures a single wavelength region at a time (serial data collection) and the relatively large monochromator cannot be placed very close to the specimen. Rarely used in TEM; sometimes used in SEM where there is more room in the specimen chamber to place a large detector.
  2. Energy Dispersive Detector (EDS) -- Emitted X-rays strike a solid state detector producing a transient electrical signal who's energy is proportional to the energy of the X-ray. These signals are sorted according to their energy and counted in a multichannel analyzer. The X-ray detector is relatively small, can be placed fairly close to the specimen and collects data in parallel fashion (i.e. it collects all wavelengths at once); thus, this is by far the most commonly used method. The energy resolution is only 150-200 eV, but this is still good enough for semi-quantitative analytical work. Nevertheless, EDS still only collects a fraction of the X-rays. Also, X-ray fluorescence is a secondary event and its intensity depends upon the fluorescence yield of the event which can be quite low. Consequently, the efficiency of data recording is typically only 1-2 % vs. 20-50% for EELS.
3. Minimum detectable mass----10-19 gm = 5000 carbon atoms.

D. Methods of Analysis

1. CTEM mode

  1. Focus incident electron beam on an area of interest (e.g. cell organelle) and collect information on elemental composition via either EDS or EELS. Then move the beam to another area and repeat. This is a crude but useful form of imaging with a resolution determined by the size of the electron beam.
  2. Separate electrons by energy and form a CTEM image (darkfield) using normal electron optics

  3. ===> EELS (energy filtered) imaging. Fig. 9b.
2. SEM and STEM mode--measure part or all of the EDS or EELS spectrum at each pixel of the STEM image. Display on a CRT the signal arising from X-rays or electrons of specific energy ==> X-ray mapping.

V. Radiation Damage

A. Signal-to-Noise for High Resolution Imaging

1. In the ideal case, the S/N is determined by the statistics of electron counting; the signal is the number of electrons scattered from a particular resolution element (pixel) while the noise is the square root of the number of incident electrons for each resolution element (the statistical variation in number of incident electrons for each pixel). Thus, in principle any resolution (up to the instrumental limitation of 2-4 Å) can be achieved for any specimen if it is imaged with enough electrons.

2. Contrast is low for biological specimens because they are composed of relative light elements which scatter electrons weakly. Thus, to produce a 10 Å resolution of a typical biological specimen, one must image it with 250 electrons/Å2 if total contrast is 2%.

B. How Sensitive Are Biological Specimens?

1. Specimens lose 10-30% of their mass during one exposure (10-100 el/Å2).

2. Protein crystals disordered after an exposure of 1-5 el/Å2 at 0°C.

3. Enzyme activity is lost after an exposure of 0.05 el/Å2.

4. Thus, it would appear that one cannot image biological specimens at high resolution because images produced with a low enough exposure to prevent severe radiation damage would have a S/N << 1.0. The situation is much worse in the case of analytical imaging since many more electrons are required to form an image (mean free path for high energy loss event is large). But there are some limited solutions to this dilemma described below.

VI. Specimen Preparation

A. Goals--most techniques achieve only some of these goals.

1. Make thin specimens which can exist in vacuum; this usually means remove all H2O. Thin specimens are required in order to:

  1. prevent absorption of electron beam
  2. reduce multiple scattering
  3. decrease chromatic aberration from inelastic electron scattering.
2. Increase contrast--heavy atom stain

3. Decrease radiation damage

  1. heavy atom stain distribution less sensitive to radiation
  2. less damage at low tempertures
4. Render compatible with high vacuum; usually requires removing H2O

B. Thin Sectioning--tissues, cells, organelles, membranes -- most common technique in biological electron microscopy

1. Fixation--crosslink components with e.g. glutaraldehyde, formaldehyde, OsO4, uranyl acetate etc. (stain)

2. Dehydrate--replace H2O slowly by transferring through a series of solvent mixtures containing increasing concentrations of organic solvent (acetone, alcohol, ethylene glycol, propylene oxide)

3. Embed in plastic monomer--transfer the specimen through increasing concentrations of plastic monomer; then initiate polymerization by heating or with UV light.

4. Slice thin sections--100-1000 Å--may poststain at this stage with lead citrate etc.

C. Freeze-Fracture/Replication--tissues, cells, organelles, membranes

1. Freeze--with or without fixation and/or cryoprotectant

2. Fracture--expose inner surfaces by breaking or chipping away frozen layers [at high vacuum].

3. Replicate

  1. shadow with heavy atoms (e.g. Pt, W, Ta) [done under
  2. stabilize with carbon layer [high vacuum
  3. digest away specimen (bleach, chromic acid)
  4. examine replica in EM
4. Variations
  1. etching--sublime H2O away before replication
  2. freeze/dry and shadow--adsorb the specimen to a carbon film or other clean surface, freeze in thin layer of ice, sublime ice away (under vacuum) and shadow with heavy atoms.
  3. shadowing nucleic acids -- no freezing; molecules are spread out on H2O surface and picked up with electron microscope grid.
D. Negative Stain--particles (enzymes, viruses), membrane fragments.

1. Adsorb specimen to support film--thin carbon film over electron microscope grid.

2. Rinse with heavy atom solution--e.g. 1-2% uranyl acetate, uranyl formate, or phosphotungstate--draw off XS and dry.

3. As it dries, the stain forms a glassy electron dense replica of specimen. Stain-excluding areas (volumes) appear light in the image

E. Unstained Specimens--how to preserve them in vacuum in order to obtain images of the actual specimen rather than the distribution of stain around the specimen

1. Environmental stage--maintain H2O vapor around specimen by differential pumping; isolate specimen area with small apertures above and below so higher concentration of H2O can be maintained around it. Seldom used.

2. Freeze-dry or critical point dry--to remove H2O without totally disrupting the structure.

3. Embed in glucose (sugar) syrup by drying in thin layer of 1% glucose (as when negatively staining).

  1. -OH's of sugar mimic polar H2O
  2. supports specimen from collapse during drying
  3. reduces contrast--glucose has density similar to protein. One alternative is to use heavy atom-substituted sugars such as auro-thio-glucose.
4. Frozen-hydrated specimens -- best method
  1. freeze specimen in thin layer of vitreous (non crystalline) ice
  2. "natural" aqueous environment
  3. ice shows protein in positive contrast
  4. can irradiate with 5-10x more electrons at low temp
  5. requires special stages only recently available
VII. Image Processing--How can it help?

A. Image Restoration--correct aberrations of the phase contrast transfer function (sometimes called the "point spread function" of the imaging system). Most easily accomplished by operation on the Fourier transform.

B. Signal Averaging--improve low signal-to-noise ratio by adding images of identical structures together.

1. Low exposures required to prevent radiation damage in unstained specimens result in poor S/N in the image

2. Average unit cells in a crystal--Electron Crystallography (Fig. 10)

  1. Digitize and calculate the Fourier transform of the image which is non-zero only at points of a Reciprocal Lattice for perfect crystals. Spacings of the reciprocal lattice are the inverse of those of the crystal lattice.
  2. Reconstruct average image of all unit cells by inverse Fourier transformation (Fourier synthesis) using only information at Reciprocal Lattice points (see handout from previous lect.).
3. Average individual particles by Correlation Methods and Correspondence Analysis. Correspondence Analysis sorts images of individual particles into groups which are similar to one another. Images of individual particles which are similar in structure can be averaged together using Correlation methods to bring them all into a common alignment.

C. Three-Dimensional Reconstruction

1. Sorts out overlapping details in two-dimensional images

2. Tilt specimen in electron microscope--but usually only by ±60° -- like CAT scan and NMR imaging; obtain projections of structure at different angles.

3. Central Section Theorem--the 2-dimensional Fourier transform of a projection of a 3-dimensional object is a Central Section (a section passing through the origin) of the 3-dimensional Fourier transform.

  1. Thus, one can determine the 3-dimensional Fourier transform by sampling it on central sections obtained from 2-dimensional images of tilted specimens.
  2. With enough central sections, the entire 3-dimensional transform can be determined to a limited resolution (determined by the number of central sections, the more central sections the higher the resolution). The 3-dimensional structure can be calculated by an inverse Fourier transform operation on the estimated 3-dimensional Fourier transform.
4. Again, Fourier methods are especially applicable to periodic objects because their transforms have special properties. Fig. 10.
  1. Fourier transforms of 2-dimensional crystals (crystals which are periodic in two dimensions but only one unit cell thick in the third) are sampled on x,y plains but continuous along the non-periodic z-axis.
  2. Specimens with helical symmetry have Fourier transforms which are non-zero only along layerlines oriented perpendicular to the helix axis. If one knows the nature of the helical symmetry, many Fourier transform central sections can be determined from a single image ==> 3-dimensional reconstruction to limited resolution from a single images.
  3. Icosahedral Particles (spherical viruses) have Fourier Transforms with special properties
5. Other 3-dimensional reconstruction algorithms have and are being used: most common is Filtered Backprojection -- used in some CT scanners.


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