Hammond C. The Basics of Crystallography and Diffraction

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284 Electron diffraction and its applications

Incident beam

Distribution of

inelastic scattered

intensity within

specimen

Intensity distribution

around incident

beam

Fig. 11.8. The distribution of inelastic scattered intensity within a specimen and (below) the

distribution intensity around the incident (unscattered) electron beam.

than the Bragg angle (Fig. 11.9(a)). No reﬂection of the incident beam, or given the

extension of the reciprocal lattice point (Section 11.2), only a weak reﬂection of the

incident beam occurs. However, those inelastically scattered beams which are incident

to the planes at the Bragg angle are reﬂected. Figure 11.9(a) shows two such beams, AB

and AC, AB being more intense than AC since the inelastic scattering angle is smaller.

Beam AB is reﬂected to D (solid line), giving an increase in intensity; at the same time it

is attenuated in the direction to E (dotted line) giving a reduction in intensity. A similar

reﬂection (solid line) and attenuation (dotted line) occurs for AC. However, because

beam AC is weaker the overall effect will still be an increase in intensity at D and a

decrease in intensity at E. Now these peaks are not spots but are sections through cones

of intensity which intersect the reﬂecting sphere to give nearly straight lines running

parallel to the reﬂecting planes, i.e. perpendicular to the plane of the paper. These—the

‘light’ and ‘dark’ Kikuchi lines—are shown in the plane of the diffraction pattern in

Fig. 11.9(b). Also shown is the (weak) hkl diffraction spot and the downwards projected

trace of the planes which lies half-way in between the Kikuchi lines (dashed line). The

angle between the Kikuchi lines is 2θ and thus they have the same separation R as the

distance between the centre spot and hkl diffraction spot.

From Fig. 11.9(a) two angles are of interest; φ, the deviation of the planes from the

exact Bragg angle, and ρ, the deviation of the planes from exact parallelism with the

incident beam. There angles may be determined by measuring the distance a

(from the

dark Kikuchi line to the centre spot) and a

(from the trace of the planes and the centre

spot). Given the camera length L (Fig. 11.4), then φ = a

/L and ρ = a

/L.

The geometry may also be represented in terms of the Ewald reﬂecting sphere con-

struction, Fig. 11.10. k and k

are the vectors representing the directions of the diffracted

11.5 Kikuchi and electron backscattered diffraction patterns 285

Light

Kikuchi

line

(a)

(b)

Trace of

reflecting

planes

Incident beam

hkl

Dark

Kikuchi

line

Fig. 11.9. (a) A crystal with a set of planes at an angle to the incident beam slightly smaller than the

Bragg angle. AB and AC show two inelastically scattered beams (see Fig. 11.8) which are incident to

the planes at the Bragg angle and which are partly reﬂected (solid lines) and partly attenuated (dotted

lines) resulting in light (electron excess) and dark (electron deﬁcient) Kikuchi lines, shown in plan in

(b) together with the diffraction and centre spots and the projected trace of the reﬂecting planes.

and direct beams and g

hkl



≡ d

∗

hkl



is the reciprocal lattice vector for the reﬂecting planes

(see footnote, page 198, Section 8.3). The hkl reciprocal lattice point lies outside the

sphere and the vector s (closely parallel to k and k

) represents the deviation of g

hkl

from

the exact Bragg condition, the modulus of which is given by

= φ

hkl

. Now we see

the value of Kikuchi lines in establishing the precise orientation of the reﬂecting planes:

as we (say) rotate the crystal clockwise, the pair of Kikuchi lines move to the left and at

the exact Bragg angle

(

φ = 0

)

, the dark Kikuchi line passes through the centre spot and

the bright Kikuchi line passes through the diffraction spot. Similarly, when the crystal is

286 Electron diffraction and its applications

hkl

Fig. 11.10. The Ewald reﬂecting sphere construction for a set of hkl planes at an angle to the incident

beam (vector k

) slightly smaller than the Bragg angle. The reciprocal lattice vector g

hkl

lies outside

the sphere and the vector s represents the deviation from the exact Bragg condition. φ is the deviation

angle of the planes from the exact Bragg angle and ρ the deviation of the planes from exact parallelism

with the incident beam.

rotated such that the planes are exactly parallel to the incident beam, the Kikuchi lines

will be at equal distances each side of the centre spot.

In practice we may see many Kikuchi lines—as many pairs as there are diffraction

spots (Fig. 11.11). However, just two pairs of lines, and two diffraction spots, are

sufﬁcient to determine the orientation of the crystal with respect to the incident electron

beam direction. Figure 11.12 shows a pattern with two spots, h

and h

and

their associated Kikuchi lines; half way in between which are drawn the downwards

projected traces of the planes (dashed lines). The downwards direction of the zone axis

[¯u¯v ¯w] lies at the intersection of the traces as shown; the upwards direction [uvw] (i.e.

anti-parallel to the electron beam) lies on the opposite side of the centre spot and is found

by cross-multiplication of the indices as described in the footnote on p. 281. The angle

η between [uvw] and the centre spot, 000, is simply found by measuring the distance P,

again using the relationship η = P/L.

The pattern of Kikuchi lines also reﬂects the symmetry of planes in a crystal and

‘maps’ may be created showing the patterns of lines for different crystal orientations.

Figure 11.13 shows one such map for an fcc crystal centered on [001], the four-fold

symmetry of which is clearly evident. Note that the spacings of the pairs of lines corre-

spond to the reciprocal lattice spacings; 200 and 020 passing through the centre being

the narrowest, then 220, 2

20, and so on. The value of such maps is that they provide

a means of tilting the specimen into any desired orientation: one simply ‘moves along’

a pair of lines (like railway lines!) to bring the desired zone axis to the centre of the

projection.

In this situation the (simpliﬁed) explanation of Kikuchi line contrast given above breaks down. A more

detailed analysis shows that the lines are still observed.

11.5 Kikuchi and electron backscattered diffraction patterns 287

Fig. 11.11. An electron diffraction pattern of an austenitic stainless steel (fcc) showing diffraction

spots and Kikuchi lines. (Photograph by courtesy of Dr P.M. Champness and Professor G.W. Lorimer.)

11.5.2 Electron backscattered diffraction (EBSD) patterns

in the SEM

The geometry of the formation of EBSD patterns is analogous to that of Kikuchi patterns

except that the incident electron beam is at a small angle, typically ∼20

◦

, to the specimen

surface. Only those electrons inelastically-elastically scattered from depths some 10–

20 nm below the specimen surface (depending on the specimen, incident beam energy

and angle) emerge from the specimen to be recorded on a phosphor screen and a CCD

camera. Hence EBSD is essentially a surface technique and any surface damage or

distortion must be removed by careful preparation techniques. An example of an EBSD

pattern is shown in Fig. 11.14. The orientation analysis is of course carried out by

computer software. The great value of EBSD is that it can be used to determine the

orientations of crystals in a specimen and therefore the misorientations and nature of

the boundaries between them. The beam is scanned across the specimen surface and the

crystal orientation is measured at each point—the resulting map reveals the constituent

crystal morphology, orientations and boundaries. The resolution depends of course on

the diameter of the incident beam and interaction volume—at present, under favourable

conditions, resolutions of 10–20 nm are obtainable. The crystal orientations may also

be presented in the form of a pole ﬁgure in an analogous way to the X-ray pole ﬁgures

described in Section 12.5.3. They are in fact complementary techniques: X-rays record

288 Electron diffraction and its applications

Zone axis

antiparallel to

electron beam

Trace

000

[uvw]

[u vw]

Light

Intersection

of traces, zone

axis parallel

to electron beam

Dark

Kikuchi lines h

Light

Dark

Kikuchi lines h

Fig. 11.12. Determination of the exact orientation of a crystal from a spot-Kikuchi line electron

diffraction pattern. The downwards zone axis, [¯u¯v ¯w], lies at the intersection of the projected traces of

the reﬂecting planes h

and h

. The angle η of the zone axis [uvw] to the incident beam direction

is given by η = P/L.

the reﬂections non-sequentially from one set of planes in a larger volume (area and depth)

of the specimen.

EBSD is also complementary to the Laue X-ray diffraction technique (Section 9.4);

again in the X-ray technique the X-rays penetrate into a much greater depth in the

specimen (typically 5–20 μm) compared to the 10–20 nm of EBSD.

11.6 Image formation and resolution in the TEM

TheAbbe criterion for image formation, discussed in Section 7.5, applies equally to light

and electron microscopy. Electron wavelengths are very much smaller (4 pm for 100 kV

instruments) than atomic dimensions, but atomic resolution (perhaps the ultimate goal in

microscopy) is limited by the severe spherical aberration of electron (electromagnetic)

lenses which limits numerical aperture (NA) values to the order of 0.01. Using the

equation a = 0.5λ/NA gives a limit of resolution of 2Å—a dimension of the same

order as atomic dimensions and interatomic spacings. The development of higher kV

instruments (smaller λ) and improvements to the corrections (for spherical aberration and

11.6 Image formation and resolution in the TEM 289

103

215

114

125

112

200

020

220

001

013

215

131

111

311

200

103

220

125

114

125

112

215

111

311

131

215

311

112

220

020

131

111

114

125

013

311

111

112

311

131

220

311

131

111

311

131

111

Fig. 11.13. A Kikuchi ‘map’ for an fcc crystal centered on [001]; note the four-fold symmetry and the

relative spacings of the pairs of lines. Tilting the crystal ‘along’ a pair of Kikuchi lines enables other

zone axes to be brought to the centre of the projection (Map by courtesy of Prof. M. H. Loretto.)

astigmatism) in electron lenses have reduced this limit to about 1 Å. However, to obtain

such Abbe or high resolution images is not a simple matter—neither experimentally nor

theoretically. First the specimen must be very thin (of the order 2–20 nm) in order that

the inelastic scattering processes described in Section 11.5 are negligible. Second, the

specimen must be carefully orientated such that the electron beam is precisely parallel

to a prominent zone axis—and of course there are no Kikuchi lines to help guide us!

Under such conditions several diffracted beams are excited which then pass through

the objective aperture to interfere to form the image. However, there is never an exact

one-to-one correspondence between the atomic structure in the specimen and the image,

the contrast of which depends upon the amount of defocus of the objective lens and the

fact that higher-angle reﬂections are excluded by the objective aperture and which would

otherwise reﬁne the ﬁne detail in the image. Hence, it is standard practice to compare

the observed images (obtained at different amounts of defocus) with those obtained by

computer simulation. However, despite these problems ‘high resolution’ TEM images

290 Electron diffraction and its applications

Fig. 11.14. An EBSD pattern of a β (bcc) titanium alloy, a 111 zone axis is at the lower right hand

corner. (Photograph by courtesy of Dr E. Merson.)

1 nm

Fig. 11.15. Ahigh resolution TEM‘dumbbell’image of germanium; the electron beam is aligned along

a 110direction in the crystal (diamond-cubic structure). (Photograph by courtesy of TEAM, a collabo-

rative programme between Argonne National Laboratory, CEOS, FEI Company, Frederick Seitz Mate-

rials Laboratory, Lawrence Berkeley National Laboratory’s National Center for Electron Microscopy,

Oak Ridge National Laboratory. TEAM homepage: http://ncem.lbl.gov/TEAM-project/index.html.

are being routinely obtained in which the peaks of intensity clearly correspond to atom

positions. Figure 11.15 shows such a high resolution TEM of germanium which has the

diamond cubic structure (Fig. 1.36(a)). The electron beam is aligned precisely along

a 110 direction in the crystal and in this orientation the germanium atoms appear to

be arranged in pairs—the so-called ‘dumbbell’ structure. So, here we see atoms: how

delighted Robert Hooke would have been to actually see his ‘bullets’ (Fig. 1.1 in this

book).

11.6 Image formation and resolution in the TEM 291

There are, however, other, no less important, modes of image formation available

in the electron microscope which occur when only one beam—either the direct beam

(bright ﬁeld mode) or one diffracted beam (dark ﬁeld mode) passes through an objective

aperture of small numerical aperture. In the bright ﬁeld mode the contrast arises from

the varying attenuation of the direct beam through different regions of the specimen; the

attenuation arisingas a result eitherof inelastic or ofelastic scattering (or moregenerallya

combination of the two). The former case gives rise to mass-thickness contrast because

the amount of inelastic scattering depends on the thickness of the specimen and the

variations in atomic number and density of the various atomic species present. The latter

case gives rise to diffraction contrast which arises from variations in elastic scattering,

or Bragg reﬂection, of the incident beam. The subject is not simple because, as noted on

p. 206 (Section 9.1) we have to take account of the dynamical interactions between the

direct and reﬂected, re-reﬂected, etc. beams. The result of these interactions is a ‘to-and-

fro’ or pendulum-type swing of energy between the direct and diffracted beams as they

progress down through the specimen, the periodicity of which is called the extinction

distance. Hence the tapered edges of specimens, or inclined grain boundaries, show light

and dark ‘thickness’ fringes corresponding to the depths at which (in bright ﬁeld mode)

the direct beam has a maximum or minimum intensity.

Diffractioncontrast may be illustrated by two simple examples. Figure 11.16(a)shows

a thin metal specimen which is slightly bent or distorted—a very common situation.

Hence the planes across the surface are at slightly different angles to the incident beam:

when they are close to, or at, the exact Bragg angle, reﬂection occurs and hence in

these regions the direct beam is attenuated. When the direct beam is used to form the

image (bright ﬁeldmode) these regions result in diffuse, darklines called Bragg contours.

Correspondingly, when the diffracted beam is used to form the image the Bragg contours

appear bright. The contrast from crystal defects arises in a similar way. For example,

Fig. 11.16(b) shows an edge dislocation in a specimen; note that around the ‘extra’ half

plane the planes are bent or curved, the distortion decreasing further away from the

Bragg

contour

Dislocation

image

Incident beam(a) (b) Incident beam

Fig. 11.16. Diffraction contrast image formation. The origin of (a) Bragg contours in a slightly-bent

metal thin foil and (b) from the bending of planes around an edge dislocation.

292 Electron diffraction and its applications

dislocation. If the curvature is such as to bring the planes to the Bragg angle (as is shown

in Fig. 11.16(b)), reﬂection of the incident beam will occur and thus the dislocation is

imaged. Note that we do not image the dislocation as such, we only image, or record,

the bending of the planes one side, or the other, of the dislocation.

Exercises

11.1 Using de Broglie’s equation, derive a simple equation for the (non-relativistically corrected)

wavelength λ of electrons in the electron microscope in terms of m, e (electron mass

and charge) and V the microscope accelerating voltage. Given the following values of

m (rest mass) = 9.11 × 10

−31

kg, e (electron charge) = 1.6 × 10

−9

C, and h (Planck

constant) = 6.63 ×10

−34

Js, determine λ for V = 10 kV, l00 kV and 1 MV.

11.2 Figure 11.17 is an electron diffraction pattern of a thin crystal of aluminium (fcc, a =

0.4049 nm (4.049 Å)) taken using an electron microscope operating at l00 kV, and shows

diffraction spots from the ZOLZ (zero order Laue zone) only. Given theelectron wavelength

λ = 0.0037 nm (0.037 Å) and the camera length L = 688 mm, index the diffraction

spots and determine the zone axis of the diffraction pattern. (Note: In face-centred lattices,

reﬂections from the planes for which h, k and l are not all odd or all even integers are

forbidden.)

(Hint: Refer to Table A.6.2 listing the indices {hkl}and d-spacings of the allowed reﬂecting

planes. Determine the d

hkl

-spacings of the diffraction spots in the pattern (see Section 11.3)

Fig. 11.17. An electron diffraction pattern (ZOLZ) of a fcc crystal. The camera length L (adjusted for

the print size) is 688 mm (see Exercise 11.2).

Exercises 293

and hence, using the table, ﬁnd their indices {hkl}. Using the addition rule in Section 6.5.2,

assign a consistent set of indices hkl to the diffraction spots and hence ﬁnd their common

zone axis using the equation in Section 6.5.7.)

11.3 In an electron diffraction experiment the bcc crystal described in Exercises 9.1 and 9.2 is

irradiated in the [100] (or a

∗

) direction with an electron beam of wavelength 0.002 nm

(0.02 Å). Find the reciprocal lattice points which lie close to the sphere in the ZOLZ (zero

order Laue zone) and FOLZ (ﬁrst order Laue zone).

(Hint: Using the same reciprocal lattice scale as in Exercises 9.1 and 9.2, it will be found

that a very large sheet of paper and very large compasses are needed to draw the complete

reﬂecting sphere! However, a sketch of the surface of the sphere near the origin will sufﬁce.

In electron diffraction, sections of the reciprocal lattice perpendicular to the electron beam

(i.e. normal to the [100] or a

∗

) direction are most convenient. Draw the section through

the origin 000 containing the 0kl reciprocal lattice points (i.e. the ZOLZ), and the section

containing the 1kl reciprocal lattice points (i.e. the FOLZ) (see also Fig. 6.8(a)). Estimate

the regions of these reciprocal lattice sections, and hence the reciprocal lattice points, which

are close to the surface of the reﬂecting sphere.)

11.4 Figure 11.18 is an electron diffraction pattern of a particle which has been extracted from

a nitrided chromium-containing steel. It is one of three possible compounds, CrN, Fe

Cor

(Cr,Fe)

.Thed

hkl

-spacings and indices of the lowest angle reﬂections are given below.

Given λL = 38.3 Å mm, identify the compound and index the diffraction pattern.

Fig. 11.18. An electron diffraction pattern (ZOLZ) of a particle extracted from a nitrided chromium-

containing steel. The camera constant λL (adjusted for print size) is 38.3 Å mm (see Exercise 11.4).

Data for CrN from the ICDD Powder Data File (11–65), for Fe

C and (Cr,Fe)

from Interpretation of

Electron Diffraction Patterns, 2nd edn, 1972 by K. W. Andrews, D. J. Dyson and S. R. Keown, Adam Hilger

(1972).