1. Convergent Beam Electron Diffraction & It’s Applications John F
Convergent Beam Electron Diffraction & It’s Applications John F. Mansfield University of Michigan Electron Microbeam Analysis Laboratory

2. Applications Examples
Outline 1
Outline
Introduction to CBED
Applications Examples

3. Applications Examples
Outline 2
Outline
Introduction to CBED
Applications Examples

4. Why CBED Why CBED? Why not SAD? Limits of Conventional SAD
Conventional SAD uses an aperture to define the area from which the pattern is to be recorded. The aperture is placed in the image plane of the objective lens to create a virtual aperture in the specimen plane (Le Poole 1947). The spatial resolution in SAD is limited by both spherical aberration and the ability of the operator to focus the aperture of the and the image in the same plane. The error in area selection U is given by:
U=Cs(2qB)3+D2qB
where: Cs= spherical aberration coefficient
qB= Bragg angle
D= minimum focus step.
The result is that the theoretical lower limit of area
selection is ~0.5µm (in practice governed by aperture
size).
Why CBED
J.B. Le Poole, Philips Tech. Rundsch 9 (1947) 33.

5. SAD Example Silicon <100> zone axis pattern. Recorded with
Gatan 622 wide-angle
CCD TV camera

6. CBED Example M X <111> zone 23 6 axis pattern. Recorded
axis pattern. Recorded
at 200kV with Gatan 622
wide-angle TV camera.

7. More information than simple spot patterns
CBED Discs.
More information than simple spot patterns
Points in discs

8. How to obtain CBED patterns
How to obtain a CBED pattern
1. Focus image at eucentric height.
2. Excite C1 to yield a small spot size.
3. Focus probe with C2 on to the area of interest.
4. Press Diffraction button.
5. Optimize pattern.

9. Even poor looking patterns can be useful!
Poor Pattern Example
Even poor looking patterns can be useful!
[1123] Laves zone axis pattern.
Even poor quality ZAPs are often
useful in phase identification

10. Learning the CBED Technique.
Learning CBED
Begin with a straightforward problem, a relatively large particle (~1-5µm) or a larger grain of material .
Learn to tilt around reciprocal space and recognize the Kikuchi lines. Try and navigate around by viewing the shadow image to save constant switching from diffraction to image and back again.
Try and use a tilt-rotate holder. It may be initially more difficult to use than a double-tilt holder, but it allows access to a greater area of reciprocal space.
Take lots of pictures. Compared with the time it will take you to get the zone axes that you need film is cheap (even grad student time!).

11. Sample, when initially placed into microscope, is usually at some random orientation.
Sample Orientation

12. Viewing the shadow image in CBED mode1. 2. 3. 4.
1. C2 heavily under-focussed.
2. C2 approaching focus.
3. C2 at focus (CBED Pattern).
4. C2 over-overfocussed.

13. Final Pattern Centering
Final pattern centering is usually performed by moving the C2 aperture a small distance. This effectively tilts the beam through the desired small angle.

14. Cr <111> XZAP <111> Cr ZAP
Convergent Beam Electron Diffraction (CBED) Pattern recorded at 120kV.
The ring pattern and bright spot at the centre of the direct disc are characteristic of a near critical voltage.
Pattern symmetry is 6mm (projection symmetry).

15. X-ray Ewald
X-ray Ewald Sphere

16. High Energy Electron Ewald Sphere

17. HOLZ Example
NbSe3 CBED Pattern with multiple HOLZ rings

18. HOLZ spacings

19. Excess/Deficit
HOLZ lines
Deficit in ZOLZ
Excess in HOLZ

20. Direct Disc Si <111> ZAP
Direct disc of a silicon <111> zone axis pattern (ZAP).
Pattern recorded at 200kV.
Sample cooled to in a liquid nitrogen cold stage to reduce the thermal diffuse scattering and sharpen the higher order Laue zone (HOLZ) line detail.

21. Problems getting patterns
Contamination.
Cool specimen (liquid N2 stage).
Clean specimen (bake out, plasma etch).
Handle all items that go into the microscope vacuum with gloves, including film & plate holders.
Keep specimen in vacuum desiccator
Problems getting patterns

22. Problems? More Problems? No HOLZ lines visible.
Record long and short camera micrographs, examine for any traces of HOLZ information.
Try an alternate zone axis, where layers may be closer together.
Try a different (lower) accelerating voltage.
Examine specimen carefully for obvious defects.
Cool the specimen (reduce Debye-Waller factor).

23. Microscope Related Criteria
To do successful CBED one needs:
Microscope Related Criteria
A microscope capable of forming a small probe (~100 to ~2nm) with a large convergence angle.
Microscope design should be such that the diffraction plane can be viewed out to at least 5° without interference from the lens pole-pieces.
Sample stages that allow either tilting in 2 orthogonal directions or tilting in one direction and rotation in the plane of the specimen cup (the latter is to be preferred).
A microscope that has a very clean vacuum system, preferably ion-pumped so that the probe may be focussed on to the specimen to have the ability to vary the microscope voltage continuously.

24. Specimen Related Criteria
To do successful CBED one needs:
Specimen Related Criteria
Specimens that are, as far as possible, free from hydrocarbon contamination. Remember contamination rate proportional to 1/d2, where d is the probe diameter.
Wash specimens thoroughly to remove any kind of adhesive used in preparation.
Always handle with tweezers.
Clean with Plasma Etcher/Bake-out.

25. Applications Examples
Outline 3
Outline
Introduction to CBED
Applications Examples

26. CBED Applications Examples
Phase Identification
Symmetry Determination - point & space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter Measurement.
Structure Factor Determination.

27. CBED Applications Examples
Phase Identification
Phase Identification
Symmetry Determination - point & space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter Measurement.
Structure Factor Determination.

28. Partial ZAP Map of a Laves Phase particle from an Al-SiC Composite
Laves Partial Zap Map
[2113] __
[1012] _ _ _
[1123]
[0111]
[2023] _
[1011] _

29. Poor Quality Laves ZAP [1123] Laves zone axis pattern.
Even poor quality ZAPs are often
useful in phase identification

30. Stererogram for Laves EMAL U of M Standard Hexagonal Stereogram
Patterns in question
were recorded from
the shaded area
EMAL
U of M
Stereogram generated
by Diffract™

31. Calculated Kikuchi Map for Laves
Partial Kikuchi Map of Laves Phase
for comparison with experimental
CBED ZAP Map.
EMAL
U of M
Kikuchi Map
generated by Diffract™

32. Laves XEDS spectrum
EMAL
U of M

33. Laves Particle
0.5µm
Laves phase precipitate in the aluminium matrix of an Al-SiC composite.
Particle was thick and faulted, however, it was easily analyzable by CBED.
Laves

34. Symmetry Determination
CBED Applications Examples
Symmetry Determination
Phase Identification
Symmetry Determination - point & space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter Measurement.
Structure Factor Determination.

35. Assume semi-infinite crystal

36. Point Group Determination
[001] Er2O3
Point Group Determination
Point group 1 m m m m m m
2mm
2mm
2mm BF
ZOLZ WP

37. Pattern Symmetries Buxton et al tables
Originally from Buxton et al (1976) Phil. Trans. Roy. Soc. 281, 181.

38. Relation between the diffraction groups and the crystal point groups
Buxton et al tables 2
Originally from Buxton et al (1976) Phil. Trans. Roy. Soc. 281, 181.

39. Point Group Determination
[011] Er2O3
Point Group Determination
Point group 2 m m m m
2mm m BF
ZOLZ WP
Note the dark bar in this disc, it is important and will be discussed later.

40. Point Group Determination
[111] Er2O3
Point Group Determination
Point group 3 3 BF 6 3
ZOLZ WP

41. Point Group Determination
Er2O3
Point Group Determination
Point Group Determination
Point Group m3 _

42. Possible reasons for Symmetry to deviate from that which is expected
Symmetry Deviations
Crystal Defects - Point defects, dislocations, stacking faults
Element not in mid-plane
Glide or Screw out of surface
Probe smaller than unit cell
Heavily tilted sample

43. In thin samples these occur as missing or dim reflections.
Dynamic Absences 1
Dynamic Absences.
In thin samples these occur as missing or dim reflections.

44. Dynamic Absences 2 Dynamic Absences.
In thicker samples double diffraction puts intensity into forbidden discs.

45. Dark bars form along the centers of the forbidden discs .
Dynamic Absences 3
Dynamic Absences.
Dark bars form along the centers of the forbidden discs .

46. Dynamic Absences 4
Dark Bars Gjønnes-Moodie Lines Dynamic Absences

47. Dynamic Absences 5
Orientation of absences with respect to symmetry elements indicates whether space group operator is screw axis or glide plane.

48. What Dynamic Absences Mean
Dark Bars, Dynamic Absences
or Gjønnes-Moodie Lines
What Dynamic Absences Mean
A bright field mirror line parallel to a line of dynamic absences indicates that there is a glide plane parallel to the incident beam.
A bright field mirror line orthogonal to a line of dynamic absences indicates that there is a 21 screw axis or its equivalent perpendicular the mirror line. (The 41, 43, 61, 63 and 65 screw axes all include the 21 operation).

49. HOLZ Dark Bars
Dark Bars with double diffraction via the HOLZ

50. Space Group Determination
[011] Er2O3
Space Group Determination
Space Group
Phase determined to be body centered by analysis of the [001] ZOLZ to FOLZ spacing.
Dark bars indicate a glide parallel to the beam direction in the [011] pattern.
Examination of tables in Tanaka & Terauchi (1985) reveals the space group to be Ia3. _
033 _
011 _
Bright field mirror (from previous
pattern).
“Convergent Beam Electron Diffraction”, Tanaka & Terauchi (1985), JEOL Ltd. m

51. More on Dark Bars Dark Bars, Dynamic Absences or Gjønnes-Moodie Lines
Dynamic absences only occur along the principal axes of a zone axis pattern when the crystal is accurately aligned on a zone axis direction.
Alternate reflections along a systematic row in a given layer MUST all show the characteristic line of absence.
The lines of absence should become narrower as the thickness increases.
The absences should occur for all thicknesses and microscope accelerating voltages.
Upon satisfying the Bragg condition for any particular order that contains an absence, a second line of absence will be observed orthogonal to the first line (the “black cross”) This condition is only strictly obeyed in the ZOLZ where three dimensional diffraction is weak.

52. CBED Applications Examples
Phase Fingerprinting
Phase Identification
Symmetry Determination - point & space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter Measurement.
Structure Factor Determination.

53. Phase Fingerprinting Example: M23 X6 <110> ZAP
Certain ZAPs in alloy systems are so characteristic that they may be identified merely by comparing them to a standard "fingerprint". This phase, based on a chromium carbide, exists in a great variety of compositions, however, the pattern is always characteristic enough to identify the phase.

54. Phase Fingerprinting Atlas
Collection of ZAP, ZAP Maps and XEDS Spectra from phases seen in steels and superalloys
Phase Fingerprinting Atlas

55. Thickness Measurement
CBED Applications Examples
Thickness Measurement
Phase Identification
Symmetry Determination - point & space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter Measurement.
Structure Factor Determination.

56. Thickness Measurement Theory 1
Theoretical Considerations for
Thickness Measurement
Thickness Measurement Theory 1
Convergent beam diffraction discs are maps
of diffracted intensity as a function of incident
wave angle and therefore have a direct
correspondence to a rocking curve. In the
two-beam approximation the rocking curve  2
for the diffracted intensity 
is given by
(Hirsch et al. 1965):  2 
sin   kz
(1)
Where:  k  1  ( s  g ) 2  
tan  1 ( s  g ) s
is the deviation parameter,  g
is the
extinction distance and z
the foil thickness.
"Electron Microscopy of Thin Crystals",
Hirsch et al
(1965).

57. Thickness Measurement Theory 2
Differentiation with respect to s reveals that  2
Thickness Measurement Theory 2
the minima of 
in equation (1) obey the
relationship:  kz 
integer
(2a)
and the maxima obey the relationship:
tan   kz 
(2b)
Also, s=0 is always either a maximum or
minimum. Kelly et al. (1975) expressed
equation (2a) as: ( s i n k ) 2   1   z
(3a)
P.M. Kelly et al.,
Phys. Stat. Sol.
(1975)A31, 771. ( s i n k ) 2
It is evident that a plot of
against ( 1 n k ) 2
in a two-beam condition yields a ( 1 z ) 2
straight line with intercept
and slope 1 of ( ) 2 .
This is the basis of the CBED 
thickness measurement technique that is
now well known.

58. Thickness Measurement Theory 3
In 1981, Allen noted that the equation 3a can
Thickness Measurement Theory 3
be rewritten thus: ( s i x k ' ) 2   1   z (
3b)
Equations 3a and 3b are equations of the
same straight line and the accuracy of the
thickness measurement can be nearly
doubled by the using both sets of fringes.
The subscripts of these two equations need
to be carefully noted since the values of s i
are labeled separately for maxima and
minima.
The values of n k
are a sequence of integers
and the values of x k
are a sequence of non
integers.
S.M. Allen, Phil. Mag. (1981)A43 325 .

59. Thickness Measurement Theory Graph
Schematic of
thickness plot

60. Thickness Measurement Theory Diagram
Schematic of
measurements needed for
thickness plot

61. Thickness Measurement in NIH Image 1
EMAL
U of M
Two-beam convergent beam electron diffraction pattern acquired into a modified version of NIH-Image.
Extra code added to the application is accessed from the custom “Thickness” menu.

62. Thickness Measurement in NIH Image 2
Data entry dialog allows the user to enter the information necessary to determine the thickness.
Default supplied values are for Silicon 220 at 200kV.
EMAL
U of M

63. Thickness Measurement in NIH Image 3
It is necessary to identify the spacing of the discs by clicking at equivalent points in the 000 and g reflections.
The floating dialog “Bragg Angle Measurements” prompts the user for the next required action.
EMAL
U of M

64. Thickness Measurement in NIH Image 4
EMAL
U of M
The fringes are identified by the user clicking on each one and then clicking on the calculate button.
Both dark and bright fringes may be measured, or each set individually.

65. Thickness Measurement in NIH Image 5
When all the fringes have been entered, the calculate button is clicked.
The application plots the curve assuming that the first fringe is n=1.
EMAL
U of M

66. Thickness Measurement in NIH Image 6
The user can simple iterate through possible values of n or select one.
The best fit plot may be saved as an image.
The thickness and extinction distance determined from the plot are included on the graph.
EMAL
U of M

67. Strain & lattice Parameter Measurement
CBED Applications Examples
Strain & lattice Parameter Measurement
Phase Identification
Symmetry Determination - point & space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter Measurement.
Structure Factor Determination.

68. <001> CBED ZAPs from Si & SiGe Alloys
CBED ZAPs from Si, Si2%Ge & Si3%Ge cross section samples.
Center of direct disc. Patterns recorded at 150kV, with a beam convergence of ~8mrad. Probe diameter ~30nm. Samples were held at liquid nitrogen temperature to reduce the thermal diffuse scattering. The orthorhombic distortion of the lattice is most obvious in the Si2%Ge sample.

69. Calculated Patterns Si
Si2%Ge
Si3%Ge
CBED Simulations based on the kinematical approximation for Si, Si2%Ge
and Si3%Ge. <001> zone axis pattern, convergence angle approximately
10mrad. Voltage set to 149.5kV by comparison with the Si pattern.
Lattice parameters:
Silicon a=0.5429nm, Si2%Ge a=0.5434nm b= nm c= nm
Si3%Ge a= nm b= nm c= nm
HOLZ Plots
generated by Diffract™

70. Structure Factor Determination
CBED Applications Examples
Structure Factor Determination
Phase Identification
Symmetry Determination - point & space group.
Phase Fingerprinting.
Thickness Measurement.
Strain & Lattice Parameter Measurement.
Structure Factor Determination.

71. Structure Factor Determination 1
CBED
Crystal
The ultimate goal here is to develop a technique that allows us to determine a completely unknown crystal structure from convergent beam electron diffraction patterns

72. Structure Factor Determination 2
Experimental Zone Axis CBED patterns are compared to calculated patterns. A number of parameters are varied to minimize the differences between the real and calculated patterns.
Structure Factor Determination 2
Structure Factor Determination 2
Real CBED
Calculated CBED
Crystal Structure

73. Energy Filter Patterns
Filtered
Unfiltered
Energy Filter Patterns
Comparison of an unfiltered and zero-loss only silicon <110> zone axis pattern. Note the large amount of thermal diffuse scattering

74. Si <110> filtered pattern
Sample: Copper <110> ZAP.
Voltage: 118kV.
Liquid Nitrogen Cooled.
C2 Aperture: 200µm.
Pattern: 512x400 pixels.
Contrast enhanced.
Si <110> filtered pattern

75. Fitting Routine.
Compares the intensities calculated from the Bloch-wave coefficients of the many beam equations.
Uses a Quasi-Newton method to minimize the sum-of-squares function:
where: Iexpt is the experimental intensity, Itheory is the calculated intensity, c is a normalization coefficient, Ni is the number of data points and B is the background level.
Quasi-Newton method chosen because:
A. approximately linear scaling in time with number of parameters
B. rapid convergence, 4 or 5 iterations.

76. Fitting Parameters. A number of parameters are needed for the fit:
Crystal Structure. Known in this case. For an unknown this would need to be determined by CBED.
Lattice Parameters. Again these are known but would need to be determined for a real unknown structure.
Debye-Waller Factor. Estimated from the mean square vibrational amplitude in Radi (Acta Cryst. A26(1970) p41).
Specimen Thickness. Determined by an initial thickness scan of the data.
Normalization constant c is essentially a scaling factor for the theoretical intensities and it is determined from thickness scan.
Thickness scan calculates  for each thickness using neutral atom structure factors.

77. Fitting Variables.
Vary the low order structure factors (Ug) and fix all the others at their neutral atom values.
Starting values for the structure factors are the neutral atom values, Doyle & Turner (Acta Cryst. A24 (1968) p390) for the elastic parts and Bird and King (Acta Cryst. A46 (1968) p202) for the absorptive parts.
Sample thickness and normalization coefficient obtained from a thickness scan.
Background levels (Bn). Vary as a function of scattering angle.

78. Fit Results. 1 2 1. “Raw” data extract from CBED discs.
2. Fitted pattern.
3. Residual map
Specimen: Copper.
Thickness: 1052nm. 3

79. X-ray and Electron Structure Factors for Cu
Parameter Data Set 1 Data Set 2 Data Set 1 Data Set 2 Neutral Atom
Ug values Ug values fx values fx values
{111} 3.883(3) 3.873(6) 21.51(1) 21.53(2) 21.76
{002} 3.426(4) 3.403(6) 20.13(2) 20.18(3)
{220} 2.422(12) 2.371(19) 16.16(7) 16.41(10)
{113} 2.035(11) 2.005(17) 13.97(10) 14.15(13) 14.05
t (in nm)
-rays X-Rays Electron Diffraction
Schneider Takama Smart & Fox & Mansfield et al
et al. & Sato Humphries Fisher Thick Sample
{111} (5) 21.80(6) (4) [21.51(1)]
{200} (4) 20.28(11) (4) [20.33(2)]
{220} (5) 16.75(8) (8) [16.16(7)]
{311} (4) 14.74(4) (7) [13.97(10)]
Estimated 2 values in parentheses.
Values in brackets are low temperature and the others are room temperature.
Top table: The fx values are X-ray structure factors at near liquid nitrogen temperature derived for the fitted Ugs using the Mott formula. Neutral atom structure factors derived from Doyle and Turner.

80. Abbreviations. Abbreviations AEM Analytical Electron Microscopy
(C)TEM (Conventional) Transmission Electron Microscopy
CB(E)D(P) Convergent Beam (Electron) Diffraction (Pattern)
SAD(P) Selected Area Diffraction (Pattern)
ZAP Zone Axis Pattern
HOLZ Higher Order Laue Zone
FOLZ First Order Laue Zone (S=Second, T=Third)
C2 Second Condenser Aperture
WP Whole Pattern
CL Camera Length