diffraction patterns obtained with convergent electron beams yield more information than patterns obtained with parallel electron beams:

Transcription

1 CBED-Patterns Principle of CBED diffraction patterns obtained with convergent electron beams yield more information than patterns obtained with parallel electron beams: specimen thickness more precise information on lattice parameters crystal system and true 3D symmetry of the atom arrangement enantimorphism, if present moreover: high spatial resolution! electron diffraction with parallel beams: resolution diameter of SAD aperture CBED: resolution minimum probe diameter experimental problems: contamination CBED requires clean specimens, UHV in the microscope local specimen heating thermal expansion, thermal stresses measured lattice parameters may not correspond to bulk values ideally, use double-tilt cooling holder for the specimen 155

2 Obtaining CBED Patterns increase beam convergence by switching off C1 upper half of the objective lens forms highly convergent beams C2 aperture C2 lens upper objective lens specimen lower objective lens diffraction discs back focal plane 156

3 microscope variables for CBED beam-convergence semi-angle, C2 aperture limits for fully focused C2 ( brightness ) switch during CBED observation calibrate using pattern of known crystal thin specimen small 2 medium 2 large 2 Kossel Möllenstedt pattern Kossel pattern camera length, L (magnification of the diffraction pattern) choose depending on desired information (fine detail in the 000 disk: 1500 mm; whole view of Kossel pattern: 500 mm) 157

4 effect of decreasing the camera length: focus of the diffraction pattern adjust minimum beam diameter with C2 ( brightness ) adjust objective lens pre-field overfocus focus underfocus size of the beam (diameter) C1, spot size determines resolution (but mind beam broadening) use small beam for specimens with strong spatial variation of strain specimen thickness thin specimen kinematical electron diffraction minimum specimen thickness required for CBED (!) 158

5 Version 120 beam tilt final adjustment to make CBED pattern symmetric mechanical tilt often too coarse preferentially use dark-field tilt (DF tilt) using bright-field tilt for this purpose misaligns the microscope for bright-field imaging position of the C2 aperture often used to adjust symmetry of the CBED pattern but: misaligns the illumination system prefer beam tilt (DF) CBED in STEM Mode obtain a STEM image of the region of interest stop the beam over the point of interest 159

6 in STEM mode, the viewing screen and the STEM detector view a diffraction pattern the scanning electron beam is focused and highly convergent when stopping the beam, a CBED image appears on the screen both, the CBED pattern and image can be viewed conveniently Laue-Zones large camera length L CBED pattern SAD pattern small camera length L higher order Laue-zones 160

7 ZOLZ patterns SAD reflecting planes normal to the direction [UVW] of the primary beam: hu + kv + lw = 0 HOLZ intersection of the Ewald sphere with higher REL planes appear as HOLZ rings if convergence (C2 aperture) is small diffraction from planes that are not parallel to the primary beam FOLZ: hu + kv + lw = 1 SOLZ: hu + kv + lw = 2 and so on 161

8 radius of HOLZ ring interception of Ewald sphere with REL planes spacing of the planes normal to the direction [UVW] of the primary beam this means: CBED patterns contain 3D information. thermal vibration weakens HOLZ intensity use N 2 cooling smaller accelerating voltage increases curvature of the Ewald sphere facilitates observation of HOLZ rings Indexing HOLZ Patterns index the ZOLZ for which hu + kv + lw = 0 identify the poles of the principal planes constitution the FOLZ, where hu + kv + lw = 1 identify the poles of the principal planes constitution the SOLZ, where hu + kv + lw = 2 check for forbidden reflections 162

9 index the HOLZ maxima Examples of ZOLZ, FOLZ, and SOLZ patterns face-centered cubic crystals 163

10 body-centered cubic crystals each HOLZ pattern retains the symmetry of the [UVW] zone but: HOLZ patterns are often shifted relative to the ZOLZ pattern 164

11 Kikuchi Lines in CBED Patterns SAD patterns: diffuse Kikuchi lines CBED patterns: sharp lines CBED versus SAD: from smaller object regions ( sharper) contribution of elastic scattering where the lines cross CBED disks inelastic scattering center distribution of inelastic scatter zone axis <uvw> B incident parallel beam hkl diffraction lines thin specimen electrons within probe at exact Bragg angle to hkl plane 2 B incident convergent probe hkl diffraction lines thin specimen back focal plane 2 B Kikuchi lines back focal plane 2 B 000 Kikuchi lines nevertheless, the lines in CBED patterns are also named Kikuchi lines 165

12 HOLZ Lines Kikuchi lines also arise from inelastic scattering by HOLZ planes HOLZ Kikuchi lines more useful than ZOLZ Kikuchi lines because from planes with much larger Bragg angles (larger g ) more sensitive to changes in lattice parameter g = 1 d hkl g = d hkl ( d hkl ) 2 the smaller d hkl, the larger g at the same d hkl particularly useful: HOLZ lines elastic part of HOLZ Kikuchi lines ( part of HOLZ Kikuchi lines within CBED disks) formation mechanism (analogy with Kikuchi lines): electrons within the incident beam at Bragg angle for diffraction at a HOLZ plane scattered out to high angles bright line through the corresponding HOLZ disk corresponding intensity deficiency in the 000 disk dark line HOLZ lines occur in pairs of bright (excess) and dark (deficiency) lines 166

13 Version 120 schematic drawing of the arrangement: deficient HOLZ lines Deficient HOLZ Kikuchi lines ZOLZ Kikuchi lines ZOLZ diffraction discs example: 167

14 Indexing HOLZ lines allowed FOLZ reflection but not intercepted by Ewald sphere expanded view of 000 allowed FOLZ reflection intercepted by Ewald sphere 168

15 requires two CBED patterns: one at large L (HOLZ lines) and one at small L (HOLZ disks) each HOLZ line pair will be perpendicular to the g-vector from 000 to the HOLZ maximum there should be a parallel HOLZ deficiency line in the 000 disk this line can be assigned the same indices as the HOLZ maximum Using CBED Techniques Determination of the Foil Thickness knowledge of the local foil thickness is important for many TEM techniques (example: absolute quantification in EELS) determination of foil thickness is an important application of CBED two-beam condition (only one strong reflection hkl) disks contain parallel intensity oscillations number of maxima and minima increases as one moves the beam from thinner regions to thicker regions of the foil the oscillations are symmetric in the hkl disk and asymmetric in the 000 disk 169

16 example: intensity variation corresponds to rocking curve (variation of diffracted intensity with the direction of the incident beam) similar to bend contours: procedure to extract thickness: 170

17 in the hkl disk measure distance between the central bright fringe and each of the dark fringes the central bright fringe is at the exact Bragg condition, where s = 0 the fringe spacings correspond to angles i from these spacings on obtains the deviations s i for the i th fringe from the equation s i = i 2 2 B d hkl B : Bragg angle for the diffracting hkl plane; d hkl : spacing of the reflecting lattice planes; s: magnitude of the excitation error s. hkl is given by the separation of the centers of the two disks 171

18 if the extinction distance g is known for the reflection hkl under consideration, one can determine the foil thickness t from where n k is an integer. s i 2 n k g 2 n k 2 = 1 t 2 if g is not known, one can determine t by the following graphical method: arbitrarily assign n = 1 to the first fringe, where s = s 1 assign n = 2 to the second fringe, where s = s 2, and so on plot (s i /n k ) 2 versus (1/n k ) 2 if the result is a straight line, the assumption n = 1 for the first fringe was good, which will be true if t < g if the result is a curved line, repeat with n = 2 for the first fringe, then n = 3, n = 4, until a straight line is obtained once a straight line is obtained, the slope of the line corresponds to (1/ g ) 2 the intercept with the y-axis corresponds to (1/t) 2 Lattice Parameters fundamental translations of the crystal lattice: a, b, c SAD pattern only contains information about repeat distances normal to the direction of the primary electron beam CBED: radii of HOLZ rings repeat distance parallel to primary beam 172

19 from a single CBED patterns one can determine all the lattice parameters (a, b, c) of the unit cell relation between spacing H of the REL planes parallel to the electron beam and radii G 1 and G 2 of the two innermost HOLZ rings: G 1 = 2H H 1 = 2 G 1 2 G 2 = 2 H H 1 = 4 G 2 2 for a known crystal system, one can determine all three lattice parameters from a single CBED pattern experimental problems if HOLZ rings are split, measure the inner radius 173

20 mind distortions for large diffraction angles ( 10 ) experimentally, it may be easiest to measure HOLZ ring radii from Kossel patterns rather than from Kossel-Möllenstedt patterns Lattice Centering comparison of ZOLZ and FOLZ reflections in Kossel-Möllenstedt patterns primitive lattice: no forbidden reflections FOLZ reflections directly superimpose ZOLZ reflections face-centered cubic lattice or body centered cubic lattice: forbidden reflections FOLZ reflections shifted versus ZOLZ reflections 174

21 Point Group point group set of macroscopic symmetry elements (identity, rotation, inversion, mirror symmetry) CBED: enables direct determination of point group from two or three low-index zone-axis patterns (ZAPs) advantage over X-ray diffraction: much smaller areas unambiguous result (non-trivial in X-ray diffraction) X-ray diffraction kinematical diffraction Friedel s law: I hkl = I h k l always inversion symmetry cannot distinguish, for example, between point groups m and 2 175

22 the 32 point groups: 176

23 X-ray diffraction can only distinguish among 11 different Laue groups (obtained by adding inversion symmetry to each one of the 32 point groups) CBED: Friedel s law breaks down because of dynamical diffraction. CBED can distinguish the 32 point groups point symmetry becomes apparent in CBED zone-axis patterns (ZAPs) whole-pattern symmetry symmetry of the complete pattern, including HOLZ Kikuchi lines most important bright-field symmetry 000 disk only, when HOLZ lines are present in this case, the 000 disk also contains 3D information combination of whole-pattern and bright-field symmetry in three ZAPs usually suffices to determine the point group Point Group Determination Steed method uses several different ZAPs determine whole-pattern symmetry and bright-field symmetry in each one of them and ensure that they are consistent with the projection symmetry (diffuse intensity in the 000 disk without HOLZ lines) 177

24 for this purpose, use Buxton table of CBED pattern symmetries this table shows diffraction group (full 3D symmetry of a DP) the projection-diffraction group (full 2D symmetry) the Buxton table describes the possible diffraction groups consistent with the whole-pattern and bright-field pattern symmetry the following table describes the point groups that are consistent with the individual diffraction groups 178

25 for a sufficient number of ZAPs, there will be a unique solution for the point group Space Group Determination space group refers to the microscopic symmetry of crystal structures, thus point symmetry elements combined with microscopic translations (screw axes and glide planes, for example) 179

26 after determining (or knowing) the Bravais lattice and the point group, several space groups may still be possible CBED: analyze kinematically forbidden reflections reflections can be kinematically forbidden because the crystal lattice is centered or because the structure possesses symmetry elements like screw axes or glide planes example: glide plane because of the glide plane (dashed line), the waves reflected by the atom layers on the hkl planes destructively interfere with the waves reflected by the atom layers between theses layers the (hkl) reflection is kinematically forbidden in CBED patterns, kinematically forbidden reflections often occur because of double diffraction strong interaction between electrons and matter strong Bragg reflections ( dynamical electron diffraction ) strong Bragg reflections act as new primary beams, which are diffracted again by the crystal 180

27 possible effect in the diffraction pattern: additional reflections kinematically forbidden reflections appear for example consider g 1, g 2 REL primary reflection: S = s - s 0 = g 1 double diffraction: S = s - s = g 2 add: S + S = s - s 0 = g 1 + g 2 example: Si <110> diffraction pattern 200 reflections are kinematically forbidden nevertheless, 200 reflections appear by double diffraction: 200 = = CBED: if two or more equivalent double diffraction paths exist, the kinematically forbidden reflections will have a central line of zero intensity passing through the disk reason: diffracted beams from two equivalent paths interfere destructively in the center of the disk these dark lines in a kinematically forbidden reflection are denoted as dynamical absences or Gjønnes-Moodie lines 181

28 Gjønnes-Moodie lines usually occur along systematic rows of reflections Gjønnes-Moodie lines indicate the presence of screw aces and glide planes to ascertain whether screw axes, glide planes, or both are present, one needs to analyze the orientation of the Gjønnes-Moodie lines with respect to the BF mirrors within the 000 disk 182

29 183

30 Strain Analysis ZOLZ reflections and HOLZ-ring diameters yield lattice parameters with an accuracy of about 2 % best method: use positions of HOLZ lines in the BF disk arise from very high order reflections very sensitive to changes in lattice parameter technical procedure: record HOLZ lines in the BF disk simulate HOLZ-line pattern in the BF disk using appropriate software compare simulation with experimental data accuracy of this method in determining lattice parameters: 0.2 % focused electron beam CBED enables sensitive measurement of local lattice parameters measure spatial variation of lattice parameters owing to stresses or compositional changes limitations average over foil thickness local heating by electron beam specimen cooling advisable, but may introduce thermal stresses 184

31 surface relaxation measurement does not reflect bulk properties example: effects of compositional lattice parameter variation in Cu-Al alloys 185