electron diffraction of 2d crystals - principle and data collection

$: Electron Diffraction of 2D crystals - principle and data collection$
Yifan Cheng

Department of Biochemistry and BiophysicsUniversity of California San Francisco

Electron Diffraction of 2D crystals- principle and data collection

August 15-19, 2011TEMIMPS 2DX Workshop at UW Seattle

Electron diffraction and image of 2D crystal are two integrated parts of electron crystallography:

* Image of a 2D crystal provides both phases and amplitudes of each reflections from the crystal; However the amplitude is modified by CTF, envelope functions, ... etc. In principle, one can determine the structure of a 2D crystal using image only, with both phases and amplitudes determined directly from images. But in practice this is very difficult, particular towards atomic resolution.

Electron Crystallography of 2D crystal

* Electron diffraction provides accurate amplitudes of each reflections, but without any phase information, same as in the x-ray crystal. One cannot determine the structure of a 2D crystal only from diffraction, unless the phases can be determined by other method, such as molecular replacement, ... etc.

Outlines

1) Principle of electron diffraction;2) How to set up the microscope for diffraction;3) Collecting good diffraction patterns;4) Data processing;5) Recording medium: CCD/Image Plate/Film;6) What’s more to think about;

* In all the discussion here, we assume that your crystal is suitable for the electron diffraction: you have already got “perfect” crystals with sufficient large size, and you can make “perfect” grids using “perfect” carbon film on a “perfect” Mo grid. Under such assumption, we will be discussing how to get “perfect” electron diffraction pattern.

Outlines


Electron source

Condenser lens C1

Condenser lens C2

Condenser aperture

specimen

Objective lens

Back focal planeObjective aperture

Back image planeSelect area aperture

A simplified electron optic system in an electron microscope

Projection lens systemImage shift/diffraction shift coils

viewing screen/camera

Additional lens in the electron microscope

Electromagnetic lens

* The focal length of a electromagnetic lens can be easily adjusted by changing the lens current.* Switching between image and diffraction modes is done by pressing a button in the microscope.

Image modeClathrin coat

Image mode

* Intermediate/projection lens system was set to correspond the camera to the back-image-plan.

Diffraction mode

from bacteriorhodopsin

diffraction mode

* Intermediate/projection lens system was set to correspond the camera to the back-image-plan.

Image/diffraction formation

The image/diffraction formation in the electron microscope can be treated as two separate processes:

• The interaction of the incident beam with the specimen, described by the weak-phase object approximation, which is the theory used mostly to describe the image formation of thin specimen with light elements, such as a biological sample.

• The propagation of the electron beam from exit plane of the specimen to the back focal plan (diffraction) and back image plane (image) of the objective lens.

Weak-phase object approximation

This is a highly simplified theory based on the so-call weak-phase object, which is a very thin specimen formed mostly by low- and medium-weight molecules.

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Φ r

r ,z( )Suppose: 1) the specimen is very thin so that can be approximated by ; 2) both in-coming and exiting beams are parallel beams;

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Φ r

r ( )

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Φ r

r ( ) = Φr r ,z( )dz∫

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Φ r

r ,z( )

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ψin

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ψ'ex

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ψex '=ψ in exp iΦ(r r )( )

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ψex =ψ ineiΦ(

r r )

ψ ex ' ≈ψ in 1+ iΦ(r r )[ ]

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ψi

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Φ(r r )

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ψex

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ψin

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iΦ(r r )

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Iex = ψ ex

2= ψ in

2= Iin

Elastic scattering

Diffraction/Image formation

focal plane

image plane

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Ψbf (r k )€

ψ'ex

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ψim

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Ψbf (r k ) = F ψ 'ex (

r r )[ ]At back focal plane:

At exit plane of specimen:

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ψ'ex (r r ) ≈ψ in 1+ iΦ(

r r )[ ]

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ψim (r r ) = F−1 Ψbf (

r k )[ ]At back image plane:

At back focal plan:

The plane wave ψ’ of exit-beam travel through objective lens to the back focal plane. The wave function at back focal plane of the objective lens is the Fourier transform of the exit wave:

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Ψbf (r k ) = F(ψ 'ex (

r r )) = F 1+ iΦ(

r r )[ ] = δ(

r k ) + iF(Φ(

r r )) (7)

However the lens aberration and defocusing generate an extra phase shift to the scattered beam:

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γ(r k ) = 2πχ

r k

χ (k,ϕ ) =1

2λ Δz +

1

2sin2(ϕ −ϕ 0)

⎡ ⎣ ⎢

⎤ ⎦ ⎥k

2 +1

2λ3Csk

4(8)

Together with the aperture function A(k) the wave function at back focal plane will become:

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Ψbf (r k ) = F(ψ 'ex )A(

r k )exp(2πiχ

r k ) (9)

Electron Diffraction

At back focal plane:

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r k )exp(2πiχ

r k )

Intensity at back focal plane (diffraction intensity) is:

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I = Ψbf

r k ( )

2

= F(ψ 'ex )2

A(r k )

2

exp(2πiχr k )

2

= F(ψ 'ex )2

Therefore, the diffraction intensity is not affected by the CTF. CTF cannot be measured directly from diffraction.

In case of a crystal (or 2D crystal):

where is the structure factor (amplitude) of reflection (h,k) from a 2D crystal.

Together with the aperture function A(k) the wave function at back focal plane will become:

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r k )exp(2πiχ

r k ) (9)

Then, the wave function in the back image plane of the lens is the reverse Fourier transform of the wave function at back focal plane (⊗ is for convolution):

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ψim (r r ) = F−1{F(ψ 'ex )A(

r k )exp(2πiχ

r k )}

=1+ iΦ(−r r )⊗ J0(

r r )⊗F−1[exp(2πiχk)] (10)

The observed intensity in the image is then:

(11)

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=1+ 2Φ(−r r )⊗ J0(

r r )⊗F−1[sin(2πχk)]

What about image?

Fourier transform provide amplitude

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F(Ii(r r )) = δ(

r k ) + 2F(Φ(−

r r ))sin(2πχk)

Fourier transform of the image also gives amplitude as well as phases of a 2D crystal. Both are modulated by the CTF, it is easier to correct for the modulation to the phase (phase flipping), but less accurate for the amplitude correction.

In electron diffraction:

Intensity is the square of structure factor, without CTF modification. But there is no phase information, thus not sensitive to the specimen movement (drift).

Collecting diffraction pattern is easier than collecting image

Diffraction pattern does not contain phase information, thus it is insensitive to any instability of stage/specimen.

Taylor and Glaeser (2008) “Retrospective on the early development of cryoelectron microscopy of macromolecules and a prospective on opportunities for the future” Journal of Structural Biology

“Probably the best electron diffraction pattern of frozen hydrated catalase obtained in 1970’s.”

Outlines


SEARCH

Beam shift

Image shift

FOCUS EXPOSURE

Electron optics of Low-Dose imaging

Setup for diffraction

* Using LowDose procedure. Over focused diffraction for SEARCH, focused diffraction with parallel incident beam for diffraction.

* To maintain a parallel beam, the choice of sizes of incident beam on specimen is limited, but can be changed by adjusting Spot Size and Condenser Aperture size.


* Locate an area where there is no crystal to destroy.



* Correlate the SEARCH mode and EXPOSURE mode.



* Move a crystal into the pointer.






* Insert the Selected Area Aperture to the selected crystal.






* Record diffraction in the EXPOSURE mode.






* Back to the SEARCH mode. Move the specimen to another crystal.







Search (D)

Focus (I)

Exposure (D)

* Back to the SEARCH mode. Move the specimen to another crystal.


Electron diffraction of bR @ pH10

0°

Liquid Helium microscope @ Kyoto

University

2.5Å

A nice electron diffraction pattern

Outlines


0°

A nice electron diffraction pattern

A “perfect” electron diffraction pattern comes with:

* sharp diffraction spots towards high resolution* no mixture of spots from different crystals* symmetrical background distribution - for globe background subtraction* good symmetry: intensities of Friedel-pair are equal - good Rfriedel* spots are sufficient apart - easy for background substation* optimized exposure time to avoid over flow of central spot* ... ... ... ...

A “perfect” electron diffraction pattern comes with:

* sharp diffraction spots towards high resolutionparallel beam, well focused diffraction pattern

* no mixture of spots from different crystalsusing selected area aperture

* symmetrical background distribution - for globe background subtraction

good focus in real space* good symmetry: intensities of Friedel-pair are equal - good Rfriedel

good focus in real space, proper recording medium (CCD)

* spots are sufficient apart - for background substationselect proper camera length, using large CCD camera (8Kx8K), using detector with high sensitivity and dynamic range

* optimized exposure time to avoid over flow of central spotLowDose procedure, exposure time ...

* ... ... ... ...... ... ... ...

What do these requirements mean?

Good sample preparation

Tamir Gonen

Using beam stop

Tamir Gonen

Tamir Gonen

Exposure time

Sharp diffraction spots require parallel incident beam and diffraction focusing

Non-parallel beam results a disk for each diffraction spots

Using selected area aperture to record diffraction from the selected crystal

Sp

ecim

en

bac

k fo

cal

pla

ne

bac

k im

age

pla

ne

Symmetrical background

* background in a diffraction pattern comes from non-crystal background and electron with energy lose.



* background is described as symmetrical background.




* and non-symmetrical background, which is influenced by the focus of objective lens.




* non-symmetrical background influences the Rfriedel value.

* and non-symmetrical background, which is influenced by the focus of objective lens.


* This is more important for tilted specimen, such as this one from 60 degree tilted specimen.

60°


* This is more important for tilted specimen, such as this one from 60 degree tilted specimen.

* If the image in real space is out of focus, the asymmetrical background will be very asymmetrical.

Other things to consider

* Camera length: camera length is like magnification of diffraction pattern. The larger the camera length, the wider diffraction spots separate from each other, and better for background subtraction and spot integration. (The size of camera is the limiting factor.)

* Dose: follow low dose practice to minimize radiation damage to the 2D crystals.

* Use beam-stop to block the central direct beam and prevent overflow.

* Exposure time is often depends on the size of the crystal (larger crystal using less exposure time), SA aperture, and beam-stop size.

* Long exposure (high dose) better for high resolution spots but with a higher temperature factor.

Outlines


Processing electron diffraction pattern

* Four steps:

1) Determine lattice, tilt axis and tilt angles.


* Four steps:


2) Determine and subtract background.


* Four steps:



3) Intensity integration and quality assessment.

a*

b*

c*

tilt axis

reciprocal lattice lines

central

section

centralsection


* Four steps:




4) Merge diffraction patterns: 3D reconstruction.


45°


60°


70°


* Four steps:




4) Merge diffraction patterns: 3D reconstruction.

Outlines


Unlike for imaging, CCD is better than film for recording electron diffraction, because of the high dynamic range of the CCD camera. Large CCD camera is particularly better for high resolution diffraction patterns.

CCD v.s Film

Image plate offers much larger size than the CCD camera, but it is unclear if its more suitable for recording electron diffraction patterns.

Things to consider: Point spread function (MTF), DQE, dynamic range, and size of the detector.

Recording image: spatial resolution (PSF/MTF/DQE) is more important;Recording diffraction patterns: dynamic range and size of the detector is more important.

Outlines

1) Principle of electron diffraction;2) How to set up the microscope for diffraction;3) Microscope condition for data acquisition;4) Data processing;5) Recording medium: CCD/Image Plate/Film;6) What’s more to think about;

* Dose/exposure time used in collecting diffraction patterns;* Automating collection of diffraction patterns;* Collecting diffraction from small crystals;

electron diffraction of 2d crystals - principle and data collection

Documents

image plane image

electron beam

image formation

diffraction modes

focal plan diffraction

lens current

xray crystal

uw seattleelectron diffraction