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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

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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 objectivelens

specimen

lower objectivelens

diffraction discsbackfocalplane

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• 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

thinspecimen

small2

medium2

large2

Kosselpattern

KosselMöllenstedtpattern

– 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)

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· 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 (!)

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– 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

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• in STEM mode,

– the viewing screen and the STEM detector “view” a diffraction pat-tern

– 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”

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• 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…

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– radius of HOLZ ring

interception of Ewald sphere with REL planes

spacing of the planes normal to the direction [UVW] of the pri-mary beam

this means:

CBED patterns contain 3D information.

• thermal vibration weakens HOLZ intensity use N2 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

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• index the HOLZ maxima

Examples of ZOLZ, FOLZ, and SOLZ patterns

• face-centered cubic crystals

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• 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

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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

backfocalplane 000 Kikuchi lines

thinspecimen

hkldiffraction lines

electrons withinprobe at exact Braggangle to hkl plane

B

2incidentconvergentprobe

incidentparallelbeam

zone axis<uvw>

inelasticscatteringcenter

distributionof inelasticscatter

backfocalplane Kikuchi lines

thinspecimen

hkldiffraction lines

B

2 B2 B

• nevertheless, the lines in CBED patterns are also named Kikuchi lines

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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 =

1dhkl

g =dhkl

dhkl( )2

the smaller dhkl, the larger | g| at the same dhkl

• 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

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• schematic drawing of the arrangement:

deficientHOLZlines

ZOLZ diffractiondiscs

ZOLZKikuchilines

DeficientHOLZKikuchilines

• example:

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Indexing HOLZ lines

allowed FOLZreflection but notintercepted byEwald sphere

allowed FOLZreflectionintercepted byEwald sphere

expandedview of 000

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• 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 tech-niques (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

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• 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:

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– 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 si for the ith fringe from the equation

si = i

2 B dhkl2

B: Bragg angle for the diffracting hkl plane; dhkl: 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

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• if the extinction distance g is known for the reflection hkl under consid-eration, one can determine the foil thickness t from

si2

nk2 +

1

g2 nk

2 =1

t2

where nk is an integer.

• if g is not known, one can determine t by the following graphical method:

– arbitrarily assign n = 1 to the first fringe, where s = s1

– assign n = 2 to the second fringe, where s = s2, and so on

– plot (si/nk)2 versus (1/nk)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

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from a single CBED patterns one can determine all the lattice parame-ters (a, b, c) of the unit cell

• relation between spacing H of the REL planes parallel to the electron beam and radii G1 and G2 of the two innermost HOLZ rings:

G1 =

2H H 1

=2

G12

G2 = 2

H H 1

=4

G22

for a known crystal system, one can determine all three lattice parame-ters from a single CBED pattern

• experimental problems

– if HOLZ rings are split, measure the inner radius

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– 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 pat-terns

• 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

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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:

Ihkl = Ih k l

always inversion symmetry

cannot distinguish, for example, between point groups “m” and “2”

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• the 32 point groups:

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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 sym-metry (diffuse intensity in the 000 disk – without HOLZ lines)

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• 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

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• 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)

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• 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 be-cause 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 dif-fracted again by the crystal

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– possible effect in the diffraction pattern:

· additional reflections

· kinematically forbidden reflections appear

– for example consider g1, g2 REL

· primary reflection: S = s - s0 = g1

· double diffraction: S´ = s´ - s = g2

· add: S + S´ = s´ - s0 = g1 + g2

– example: Si <110> diffraction pattern

· 200 reflections are kinematically forbidden

200

200-

111- -111

111-

111- - -

000

nevertheless, 200 reflections appear by double diffraction:

200 = 111+ 1 1 1

2 00 = 1 11+ 1 1 1

• CBED:

– if two or more equivalent double diffraction paths exist, the kine-matically forbidden reflections will have a central line of zero inten-sity passing through the disk

– reason: diffracted beams from two equivalent paths interfere de-structively in the center of the disk

– these dark lines in a kinematically forbidden reflection are denoted as “dynamical absences” or “Gjønnes-Moodie lines”

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– Gjønnes-Moodie lines usually occur along systematic rows of reflec-tions

• 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 re-spect to the BF mirrors within the 000 disk

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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 soft-ware

– 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

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– surface relaxation measurement does not reflect bulk properties

• example: effects of compositional lattice parameter variation in Cu-Al alloys