GBS765 Hybrid methods
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1 GBS765 Hybrid methods Lecture 3 Contrast and image formation 10/20/14 4:37 PM
2 The lens ray diagram Magnification M = A/a = v/u and 1/u + 1/v = 1/f where f is the focal length
3 The lens ray diagram So we know how to magnify the object in the object plane But what is being magnified? i.e. what generates contrast? The contrast depends on the interaction between the object and the incident electron wave
4 How (glass) lenses work
5 Electron lenses Schematic diagram of magnetic lens Ray diagram for a lens (electron or optical) Path of an electron through a magnetic field
6 Magnification vs. Resolution What is magnification? Definition: Magnification is the ratio between the apparent size of an object and its actual size. What is resolution? Definition: Resolution is the smallest detail that can be distinguished in an image is usually given in units of distance, d (nm, Å) higher resolution means smaller d Resolving power is the theoretical ability of a particular instrument to resolve fine detail in a specimen What is the resolving power of the human eye? The naked eye can see objects of about mm A typical CCD camera has a pixel size of 15µm Photographic film can resolve about 1-2µm How much do you have to magnify an atom/a virus to see it with the naked eye/a CCD camera? No point magnifying beyond the resolution limit!
7 What is contrast? Contrast is what allows us to see the object that we are looking for" " "(i.e. doesn t matter how high your resolving power is if you can t see the specimen)" " "Percent contrast = 100 x I i I b / I b " "However, the ability to distinguish the signal from the background depends on the Signal-to-noise ratio:" " "SNR rms = [Var(Signal) / Var(noise)] 1/2 " " "How is contrast generated? " " " " in light microscopy, photons are absorbed by the sample, giving rise to a pattern of light and dark" " " " in EM there is no absorption, but electrons are scattered (scattering contrast)" " " " in addition, interference between scattered waves can lead to contrast (=phase contrast) " "
8 What is contrast? Contrast is the difference in intensity of the object of interest and the background Contrast needs to be >5-10% to be detectable by eye computers can detect arbitrarily small differences in contrast Whether the signal is detectable depends on the noise
9 Electron scattering and diffraction Scattering and diffraction Interaction of electrons with the specimen Scattering of electrons e- Electrons interact with the combined electric field of the nucleus and the electrons Electrons Diffraction of as X-rays: waves: + - Secondary electrons X-rays Incident beam Backscattered electrons Auger 2ry electrons Incident wave Secondary scattered (=diffracted) wave Inelastically scattered Elastically scattered Wavelength X-rays interact of only electrons with the (nm) electrons Note λ that 1.22/ E also electrons can be described where as a wave E is with acceleration wavelength voltage! 1.22/ (ev) E and also undergo diffraction phenomena. Unscattered beam
10 Electron scattering and diffraction Scattering and diffraction Scattering of electrons - e- + Scattering and diffraction Electrons interact with the combined electric field of the nucleus and the electrons Electrons Diffraction of as X-rays: waves: Incident wave Scattered wave Interference of scattered waves: diffraction Incident wave Secondary scattered (=diffracted) wave Wavelength X-rays interact of only electrons with the (nm) electrons Note λ that 1.22/ E also electrons can be described where as a wave E is with acceleration wavelength voltage! 1.22/ (ev) E and also undergo diffraction phenomena.
11 Types of contrast in TEM The TEM image is produced mainly by the elastically scattered electrons. These give rise to contrast by two main mechanisms: Scattering contrast (mass-thickness contrast) Due to elastic interactions with atomic potential field Dependent on mass thickness differences in specimen. Generated by the loss of scattered electrons (captured by the aperture) Mass-thickness differences are introduced/enhanced by the addition of heavy atom stains Phase contrast Generated by the interaction of coherently scattered and unscattered waves Same principle as diffraction Also depends on mass thickness differences Most important form of contrast in unstained biological specimens Note: Inelastic electrons also give rise to scattering contrast, but are focused in a different plane due to different energy.
12 Scattering cross-section simulated paths of electrons" through 100nm Cu and Au" Note: more scattering for " higher Z (Au)" Scattering depends on: sample thickness atom number (Z) acceleration voltage Mean free path as " Fuinction of accelerating" Voltage" Higher voltage: better" penetration" See simulation at
13 Mass-thickness contrast Areas of greater thickness or higher density scatter more electrons
14 TEM mass-thickness contrast" Areas of higher mass thickness scatter electrons more than others" " Some of these scattered electrons are captured by the aperture and lost from the beam path" " Areas of higher mass thickness will therefore appear dark in the image" " This is known as mass thickness contrast, scattering contrast, aperure contrast or amplitude contrast! "
15 Mass-thickness contrast
16 Biological samples and contrast Biological material consists mostly of low Z atoms (C, N, O) that scatter electrons to a similar extent Thickness and density differences (which give rise to massthickness/scattering contrast) are also quite small Therefore, scattering contrast in biological specimens is very low Solutions: 1. Introduce more highly scattering (high Z) atoms, such as Au, U or W 2. Use phase contrast Most common method for particulate samples: negative staining Sample is embedded in a heavy atom staining solution and air dried. Results in a heavy atom staining pattern that reflects object structure
17 Staining of biological specimens Staining increases the image constrast by introducing heavy atoms (higher Z) that scatter electrons more strongly Negative staining results in a heavy atom staining pattern that is excluded by and reflects the object structure A" B" C" D" A: Positive staining Specimen will appear dark against light background B: Negative staining Specimen will appear light against dark background C: Incomplete negative staining D: Flattening due to dehydration
18 Negative staining and scattering contrast Negative staining results in a heavy atom staining pattern that reflects the object structure Heavy atoms give rise to scattering contrast due to loss of highly scattered electrons outside aperture phase contrast is also important at high resolution Solves many problems: 1. Water: gone 2. Radiation damage: stain is radiation resistant 3. Contrast: heavy atoms introduce scattering contrast A" B" C" D" lens BFP image
19 Negative staining of bacteriophages virions procapsids P2 (stained with 1% uranyl acetate) P4 Chang et al 2008 Virology 370,
20 Effect of aperture No Ap. 40µm Smaller aperture gives higher contrast because more electrons are removed from the beam. Lower kv also gives higher contrast
21 Types of contrast in TEM The TEM image is produced mainly by the elastically scattered electrons. These give rise to contrast by two main mechanisms: Scattering contrast (mass-thickness contrast) Due to elastic interactions with atomic potential field Dependent on mass thickness differences in specimen. Generated by the loss of scattered electrons (captured by the aperture) Mass-thickness differences are introduced/enhanced by the addition of heavy atom stains Phase contrast Generated by the interaction of coherently scattered and unscattered waves Same principle as diffraction Also depends on mass thickness differences Most important form of contrast in unstained biological specimens Note: Inelastic electrons also give rise to scattering contrast, but are focused in a different plane due to different energy.
22 Electron scattering and diffraction Scattering and diffraction When the incident wavefront hits an atom it sets up a secondary radially scattered wave. The radially scattered wave represents a small perturbation of the incident wave/ Incident wave Scattered wave Interference of scattered waves: diffraction The resultant wave can be seen as a combination of the incident wave and a π/2 phase shifted wave Scattered waves from several atoms may interact constructively or destructively to form an amplitude difference (=diffraction pattern). This is too weak to observe except for crystals. The phase distribution pattern also carries information about the specimen; however, it is not directly observable. See simulation at
23 Coherence and monochromaticity Monochromatic illumination = all radiation has the same wavelength" " " Important because electrons at different wavelength are focused in different points Coherence (spatial coherence) means that the phase of a wave is the same at different points, i.e. the wavefront arrives at the same time at different points in the x-y plane" " Lasers are coherent" " Use very small point source" Phase contrast depends on coherent illumination" For pure scattering contrast imaging, coherence is not important."
24 Condensor lenses and coherence" Tungsten filament FEG tip The source (electron gun) is not a perfect point, but has a finite size = incoherent" " By using a highly excited (strong) C1 lens and selecting a small cone, coherence is improved" " The FEG source is much smaller and more intense than the tungsten or LaB6 sources = more coherent"
25 Diffraction from a periodic specimen (double slit) Diffraction of light from a double slit Phase difference 0 Waves reinforce Phase difference!/2 Waves cancel out Phase difference! Waves reinforce Angles θ at which waves reinforce are given by Bragg s law: nλ = 2d sin θ Phase difference 3!/2 Waves cancel out See simulation at
26 Scattering angle and spatial frequency Diffraction of light from a double slit Any periodic function can be mathematically described as a sum of sine waves Phase difference 0 Waves reinforce Each wave has a spatial frequency (=resolution) that corresponds to a particular spacing (ν = 1/d) Phase difference!/2 Waves cancel out NOTE: do not confuse this wave with the electron wave (with wavelength λ) λ d θ Each spatial frequency ν (=spacing d) gives rise to a wave scattered at a specific angle θ: Phase difference! Waves reinforce Phase difference 3!/2 Waves cancel out sin θ θ = λ / d = λν This is equivalent to a Fourier transform of the object function F(θ) = FT {f(x)}
27
28 φ=0 φ=90 F F= F e iφ = F (cosφ + isinφ) φ=180
29 How do we generate phase contrast? Scattering and diffraction Biological specimens scatter electrons weakly, thus amplitude differences are very small The specimen imposes a mass-thickness dependent phase shift on the incident beam, which also carries information about the specimen This phase shift cannot be observed directly In the light microscope, a phaseplate is inserted in the beam path, which adds an extra phase shift to the direct beam In EM, phase plates are difficult to make Instead an angle-dependent phase shift is imposed on the beam in the focal plane by imperfections of the lens (C s ) and defocus Incident wave Scattered wave Interference of scattered waves: diffraction Path differences in the scattered and unscattered beam leads to amplitude reinforcement at specific angles
30 Generating phase contrast Scattered waves have a 90 phase shift The phase shift is not directly observable The resultant amplitude change is very small By adding an extra 90 phase shift to the scattered waves, the phase shift is converted to an observable amplitude difference Extra phase shift is applied by defocusing the objective lens
31 How do we generate phase contrast? Biological specimens scatter electrons weakly, thus amplitude differences are very small The specimen imposes a mass-thickness dependent phase shift on the incicent beam The phase shift cannot be observed directly In the light microscope, a phaseplate is inserted in the beam path, which adds an extra phase shift to the direct beam In EM, phase plates are difficult to make Instead an angle-dependent phase shift is imposed on the beam in the focal plane by imperfections of the lens (C s ) and defocus
32 Lenses and lens aberrations A perfect lens recombines all waves scattered from a single point into a single point Lens aberrations lead to a reduction of the attainable resolution because all the scattered electrons are not focused into the same point But it also results in a relative phase shift that gives rise to phase contrast! Spherical aberration C s electrons scattered at different angles are focused in different points C s limits the attainable resolution from d 0.61 x λ / β (Rayleigh criterion) to d 0.91 x λ 3/4 x C 1/4 s! known as the point-to-point resolution (at Scherzer focus = optimal defocus) Typical value for C s = 1 4 mm Computationally, information can be recovered beyond this limit
33 Phase contrast The lens applies a phase shift χ, dependent on scattering angle θ "(#) = 2$ % ( 1 2 &f# C s# 4 ) where Cs is the spherical aberration and Δf is the defocus. λ is the electron wavelength NOTE that Δf<0 means defocus (under focus), Δf>0 is over focus The angle θ is simply related to the spatial frequency so that! "(#) = $%(&f# C s% 2 # 4 ) θ λν note that ν = 1/d, where d is a distance measure (e.g. nm) at low resolution (e.g. >1 nm), Cs has little effect How does this phase shift affect contrast?
34 The phase shift χ(ν) leads to contrast enhancements at specific resolutions ν, commonly given by sin(χ(ν)), known as the Phase Contrast Transfer Function (CTF) an additional amplitude component is given by cos(χ(θ)), the Amplitude Contrast Transfer Function The CTF is often modeled as Phase contrast "(#) = $%(&f# C s% 2 # 4 )! 2 cos("(#))+ (1!!) 2 sin("(#)) (or something similar) in which α is the fraction of amplitude contrast (typically around 5-10% for unstained cryo-em samples) What does this function look like?
35 Generating phase contrast Scattered waves have a 90 phase shift The phase shift is not directly observable The resultant amplitude change is very small By adding an extra 90 phase shift to the scattered waves, the phase shift is converted to an observable amplitude difference Extra phase shift is applied by defocusing the objective lens
36 View of CTF curve at the aperture Curve shows the degree of contrast formation at different scattering angles θ at Scherzer focus ( optimal defocus)" " "sin χ represents phase contrast" "cos χ represents amplitude contrast" "" NOTE: the angle θ corresponds to a frequency ν (nm -1 ) " and a corresponding spacing d (nm) " θ λν = λ/d Spence 2003
37
38 Contrast transfer functions ACTF for in-focus, perfect lens PCTF for in-focus, perfect lens ACTF in- focus, for lens with Cs = 4mm PCTF in-focus, Cs=4mm PCTF defocus=67.5nm, Cs=1mm (Scherzer focus) PCTF defocus 500nm, Cs=1.3mm From Slayter and Slayter
39 Effect of focus over Scherzer focus under
40 Effect of defocus Unstained cryo-em sample of HCRSV (plant virus) defocus 2.4 µm defocus 0.9 µm
41 Fourier theory of imaging Incident beam ψ 0 Specimen = ρ xyz Specimen plane ψ s ψ 0 = 1 φ xy = ρ xyz dz ψ s 1 iφ(x,y) BFP = diffraction plane Image plane ψ i ψ f ψ f = F{ψ s } = δ(0) iφ(u,v) CTF = exp[iχ(u,v)] = cosχ(u,v) + isinχ(u,v) ψ f = ψ f x CTF x A(u,v) x E(u,v) δ(0) iφ(u,v)sinχ(u,v) Strictly speaking, cosχ(u,v) also comes in here Also, let s forget about A and E for the time being ψ i = F{ψ f } = 1 iφ(x,y)*f{sinχ(u,v)} I(x,y) = ψ i 2 = [1 iφ(x,y)*f{sinχ(u,v)}] 2 1 2φ(x,y)*F{sinχ(u,v)} F(u,v) = F{I(x,y)} = δ(0) 2Φ(u,v) x sinχ(u,v)
42 CTF correction! F(u,v) = F{I(x,y)} = δ(0) 2Φ(u,v) x sinχ(u,v) This equation tells us that to an approximation the FT of the image F(u,v) is equal to the FT of the specimen projection Φ(u,v) multiplied by sinχ(u,v), where sinχ(u,v) is the CTF. Therefore, if we know the CTF, all we need to determine the specimen projection structure is to divide F(u,v) by sinχ(u,v) to get Φ(u,v). Then apply the inverse FT to Φ(u,v) to get φ(x,y), the specimen function. Right? Problem 1: first we have to figure out what sinχ(u,v) looks like Problem 2: Division by zero! Problem 3: Exponential falloff E(u,v)
43 Envelope functions Instrument-dependent Resolution limitations: 1. CTF, i.e. sinχ(ν) and cosχ(ν) 2. Aperture A(ν) 3. Exponential falloff E(ν) caused by Beam divergence Inelastic and multiple scattering Exponential falloff E(ν) = exp[ ε(ν)], where ε(ν) is an absurdly complex function of beam divergence α, C c, C s, Δf and instrument instabilities A simple version is
44 Envelope functions Normally this decay function is modeled empirically, e.g. as E(ν) = exp[ Bν 2 ] ( ) 2 ' 2 ( E (" ) = exp # $C s% 2 " 3 # $&f" + * - * ln2 - ), ( 0 * 0 2 / 2 ( V $%" 2 C 0 ) c 2 + 4/ 2 ( I 0 ) 2 + / 2 E 0 * V.exp # 0 I 0 E 0 * 2 * 4 ln2 * 2 ) 1 Beam divergence angle ( ) where B is the B factor (temperature factor) (typically B Å -2 ) The exponential decay gives rise to the information limit of the instrument (which is different from the point-to-point resolution ) / ,
45 CTF without any falloff at high res (sin χ) envelope function exp[-bv 2 ] Information limit First zero = point-to-point resolution At more defocus, the envelope function also falls off more rapidly This is especially important in biological TEM because a high defocus is used to produce phase contrast An FEG has better coherence, i.e. less beam divergence, so the envelope function is better at the same defocus
46 Fourier transform of noise
47 Measuring the contrast transfer function The CTF can be observed by looking at the FT of the image, especially in the amorphous background" " The FT is essentially equivalent to the diffraction pattern of the object multipled by the CTF"
48 Astigmatism Astigmatism is due to imperfections in the imaging system, causing the focus to vary in different directions, and resulting in an image that cannot be focused" " Astigmatism can be corrected for by magnetic or electrostatic deflectors in the microscope"
49 Resolution limitations in EM Electron wavelength λ = 0.04Å Lens imperfections C s, C c Beam divergence Defocus Δ Contrast transfer function Image = [Specimen projection] * PSF I{image} = I{specimen projection} x CTF CTF = [m cosχ(α) + (1-m)sinχ(α)] e -ε χ(α) = πλ (Δ α 2 C s λ 2 α 4 /2) ε is a function of Δ, C s, C c and beam divergence. α is the spatial frequency Amplitude contrast m < 0.1 Drift Detector imperfections Signal-to-noise ratio The specimen
50 Summary: contrast formation in EM Contrast is what makes the specimen visible by microscopy Contrast is formed primarily by the elastic interactions of the electrons with the atoms in the sample Two main mechanisms: Scattering contrast usually generated by heavy atom staining Phase contrast main mechanism in unstained biological samples (>90%) Scattering contrast is formed by the loss of highly scattered electrons outside the objective aperture Phase contrast is formed by the constructive interaction of the scattered wave and the unscattered wave and is generated mainly by defocusing (Δf) the objective lens Phase contrast is normally given as sin(!(")) where "(#) = $%(&f# C s% 2 # 4 ) The envelope function represents an exponential falloff resulting from beam divergence, current and voltage instabilities and lens imperfections and defines the ultimate resolution limit (information limit)
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