Chapter 4 Imaging. Lecture 21. d (110) Chem 793, Fall 2011, L. Ma

Transcription

1 Chapter 4 Imaging Lecture 21 d (110)

2 Imaging Imaging in the TEM Diraction Contrast in TEM Image HRTEM (High Resolution Transmission Electron Microscopy) Imaging or phase contrast imaging STEM imaging

3 a b d (100) d (010) c (a) HRTEM image o gold oil along [001] showing atomic crystal structure (b), (c) Crystal structure model

4 Multi-wall Carbon Nano Tube

5 Imaging in the TEM What is the contrast? In microscopy, contrast is the dierence in intensity between a eature o interest (Is) and its background (I0). Contrast is usually described as a raction such as: I I I s 0 C 0 I I 0 You won t see anything on the screen or on the photograph, unless the contrast rom your specimen eceeds 5-10%.

6 In summary, BF and DF images ( or TEM and STEM ) are ormed by using the transmitted and diracted beam respectively. In order to understand and control the contrast in these images, we need to know what eatures o a specimen cause scattering and what aspects o TEM operation aect the contrast. Mechanism o Contrast or TEM/STEM image 1. Mass-thickness contrast: primary contrast source o TEM/STEM image or noncrystalline materials such as polymers and biological materials. 2. Z-contrast: High resolution STEM image 3. Diraction contrast: primary contrast source o BF/DF image especially or crystalline materials such as metals. Amplitude o eit-wave contributes to the contrast. 4. Phase contrast: High/low resolution TEM image (atomic lattice image). 5. Both the amplitude and the phase contrast: are primary contrast source o electron holography image or magnetic materials 6. In some situations, image contrast may arise rom more than one mechanism and one may dominate. 7. Crystalline sample may generate mass-thickness and diraction contrast when aperture is removed.

7 Diraction contrast Diraction contrast imaging uses the coherent elastic scattering beam to orm image Diraction contrast imaging is controlled by the Bragg diraction through variation o crystal structure and orientation o specimen. Diraction contrast is simply a special orm o amplitude contrast because the scattering occurs at Bragg angle. BF and DF image is diraction contrast image by selecting the direct or diracted beam as seen igure. Beam condition o diraction contrast imaging: the incident beam must be parallel in order to give the sharp diraction spots and thereby strong diraction contrast. So we need to underocus C2 to spread the beam. Parallel Chem 793, beam Fall 2011, condition L. Ma

8 Two-beam conditions or diraction contrast imaging BF/DF image pairs along with two-beam diraction o a series o dislocation and a stacking ault in Cu- 15at%Al.

9 (200) (b) (1-11) BF (0-22) (31-1) Only two beams are strong and others are aint due to relaation o Bragg condition (s) DF (c) (a) (a) The [001] zone-ais diraction patter showing many planes diracting with equal strength. In the smaller patterns, the specimen is tilted so there are only two strong beams, the direct [000] on-ais beam and a dierent one o the {hkl} o-ais diracted beam. (b) and (c) showing the complementary BF and DF image under two-beam conditions. In Chem (b), the 793, precipitates Fall 2011, is L. Ma diracted strongly and appears dark. In (c), it appears bright Al-Li Alloy and precipitates

10 Beam condition o Imaging HRTEM (Phase contrast )Imaging Diraction Contrast Imaging

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13 HRTEM image

14 How to obtain TEM images Using the objective aperture to orm BF or DF images Using the STEM detector to orm BF and DF image HRTEM: usually without aperture To obtain meaningul TEM image Careul preparation o sample. Garbage in, and garbage out Proper setting-up the optics condition Study your materials beore TEM session Correct operation o instrument What s your question? Design the lab plan More than one session is necessary Please add more tips rom your practice, share and discuss your eperience.

15 Mathematical Preparation 1. Delta-unction and discontinuities A Dirac delta unction at a is deined by δ ( - a) and 0 or or a a - δ ( - a) d 1 It is represented by an ininitely narrow peak o ininite height at the position a, the area under the peak is equal to one.

16 The delta unction at 0, δ(), can be considered as the limit o a set o real continuous unction, such as Gaussians: δ ( ) 2 2 a lim e a a π The Fourier transorm o the constant 1 is equal to delta unction δ cδ ( ) F( 1) and dy ( ) F( c) cf( 1) c e 2πiy e 2πiy dy

17 2. Convolutions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a a R d R r g R r g r g dx X X g g dx X g X g C g δ δ the convolution with the Dirac delta unction The identity operation is we may use the vector orm For two or more dimensions * we ind that variable, By simple change o is deined as ) ( and ) ( two unctions, o the Convolution integral 1- D, In

18 Eamples o convolutions C ( ) ( ) g( ) ( X ) g( X ) dx...( a) R () T() T()*R () The high-requency details are suppressed by convolution with the wider point spread unction Convolutions are undamental to many dierent types o eperiments. For eample, a reerence signal, R(), is created and directed at the material to be investigated (such as electron beam, light or thermal pulse, etc). The sample then modiies this reerence signal, and ideally we would like to measure the modiied signal, R (). In practice, the detector system, which converts the modiied signal into an observable signal (e.g. visible light, or electron current), introduces its own ingerprint onto the signal, R (). In simple case, this would be a linear scaling o R (), but in many case, the relationship between the detected signal R () and modiied signal R () is nonlinear and can be described by a convolution o R () with a unction T(), known as the instrument point spread unction T: R ()T()*R () This means that the instrument impose a resolution limit, which results in blurring o the smallest details in the signal R, as shown in igure.

19 Fourier transorm o a one dimensional unction () is deined as F 2πiu [ ( ) ] F( u) ( ) e d Fourier inverse transorm is deined as 1 2πiu ( ) F [ F[ ( ) ]] F( u) e du

20 Fourier transorm o a 3-D unction (r) is: r is a vector F u r e 2πi u r d r I r has a coordinate (,y,z) and u has a coordinate o (u,v,w), then : F ( ) ( ) 2πi ( u+ vy+ zw u, v, w, y, z e ) ddydz

21 We may write Fourier transorm as F [ ( ) ] F( u) ( ) F e 2πiu d ( u) ( ) cos( 2πu) d + i ( ) sin( 2πu)d I the unction () is real and even unction, i.e. (-)(), the sine integral is zero, so that F ( u) ( ) cos( 2πu) 2 0 d ( ) cos( 2πu)d F(u) is a real unction

22 Property o Fourier transorms F F F F F [ ( ) ] F( u) ( ) [ ( ) ] F( u) [ ( a) ] 1 a F u a e 2πiu [ ( ) + g( ) ] F( u) + G( u) [ ( a) ] 2πiau e F( u) d

23 Multiplication and convolution Multiplication theorem the Fourier transorm o a product o two unctions is the convolution o their Fourier transorms. F F [ ( ) g( ) ] F( u) G( u) Convolution theorem : the Fourier transorm o the convolution o two unctions is the product o [ ( ) g( ) ] F( u) G( u) their Fourier transorms.

24 Fourier transorm and Diraction 1. The amplitude distribution o a very small source or the transmission through a very small aperture (slit) in 1-D may be described as δ(), or by δ(-a) when it is not at the origin. The Fourier transorm can be used to derive the diraction pattern (in general called Fraunhoer diraction pattern according to wave optics), F F [ δ ( ) ] 1 [ δ ( a) ] e 2πiua The amplitude o a diraction pattern will be proportional F(u), where u l / λ The intensity observed will be proportional to F( u) 2 1 to

25 2. Translation o an object Translation o the object in real space has eect o multiplying the amplitude in the Fourier transorm space (reciprocal space) by a comple eponential. The intensity distribution o the Fraunhoer diraction pattern is given by F(u) ^2. F[ ( a) ] F[ ( ) δ( a) ] ( ) 2πiua F u e

26 3. Rectangular aperture In the 2-D, the transmission unction o a rectangular aperture is a F (, y) Then ( u, v) ( u, v) a / 2 a / πiu b / 2 b / 2 ( πau) sin( πbv) 2 sin ( πau) 2 ( πau) sin ( πbv) 2 2πiuy sin ab πau πbv so or a diraction rom a rectangular aperture the intensity distribution is I 1 i < a / 2 and y < 0 elsewhere a b e d e dy ( πbv) 2 b / 2 The orm o the Fourier transorm o a rectangular aperture b

27 4. Circular aperture In the 2-D, the transmission unction o a circular aperture is ( ) ( i + y ) < ( a / 2), y Then 0 ( πau) elsewhere 2 πa J1 F( u) 2 πau u is a radial coordinate and J the irst order Bessel unction J 1 ( z) 0 k 2k (-1) z 2k k! ( k ) z ! k 1 2 () is the irst order Bessel unction.

28 Basic Imaging Theory 1.The imaging process As a irst approimation, we consider coherent image, i.e. a plane parallel, monochromatic electron wave alls in the object, and is scattered, the scattered waves interere and recombined by an electron lens (objective lens) to orm an image. 2. The process consists o two parts: (a). The interaction o the incident wave with the object, deining the eit wave rom the object; (b) the action o the objective lens in orming the image 3. The essential question: how is the images intensity distribution related to the object structure? 4. For detailed discussion o imaging and electron scattering, see Diraction Physics by John M. Cowley, North-Holland, Amsterdam, 1981, 2nd edition.

29 Phase contrast images Contrast in TEM images can arise due to the dierences in the electron waves scattered through a thin specimen. This phase contrast mechanism can be diicult to interpret because it is very sensitive to many actors such as specimen thickness, scattering actors, and properties o lens, etc. Phase contrast imaging can be eploited to image the atomic structure o thin specimens. the most obvious distinction between phase contrast imaging and other orms o TEM imaging is the number o beams collected by the objective aperture or an electron detector. or BF/DF image only requires a single beam selected by OA or imaging. A phase-contrast image requires the selections o more than one beam. In general, the more beams collected, the higher the resolution o the image. 7 spots are used or imaging the lattice

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31 Only two diraction spots were used or imaging the GaAS lattice showing one set o ringes rom the (111) plane (horizontal). Only (111) diraction contributed the image. The vertical crystal planes are invisible. The inormation retracted by image is not complete. Incorrect operation

32 Many beams were selected by an aperture (ring) to orm image showing more inormation about the structure o specimen. the question is where are the atoms with respect to the bright and dark contrast dots? Do the dark spots or bright spot correspond what atoms (Ga or As), etc.? Beautiul image sometimes is diicult to interpret. Correct operation

33 Beautiul image sometimes is diicult to interpret.