Theoretical aspects of TEM

5.4.4 Theoretical aspects of TEM

1 Another imaging mode does allow more than one beam to Although the examination of materials may be carried interfere in the image plane and hence crystal periodicity

5.4.4.1 Imaging and diffraction

can be observed; the larger the collection angle, which is out with the electron beam impinging on the surface at

generally limited by lens aberrations, the smaller the

a ‘glancing incidence’, most electron microscopes are periodicity that can be resolved. Interpretation of this direct aligned for the use of a transmission technique, since

imaging mode, while apparently straightforward, is still added information on the interior of the specimen may

controversial, and will not be covered here.

The characterization of materials 147 the crystal, since this information is contained in the

spacing of diffraction maxima and the directions of diffracted beams, information excluded by the objec- tive aperture.

Variations in intensity of the selected beam is the only information provided. Such a mode of imaging, carried out by selecting one beam in TEM, is unusual and the resultant images cannot be interpreted simply as high-magnification images of periodic objects. In formulating a suitable theory it is necessary to consider what factors can influence the intensity of the directly- transmitted beam and the diffracted beams. The obvi- ous factors are (1) local changes in scattering factor,

e.g. particles of heavy metal in light metal matrix, (2) local changes in thickness, (3) local changes in ori- entation of the specimen, or (4) discontinuities in the crystal planes which give rise to the diffracted beams. Fortunately, the interpretation of any intensity changes is relatively straightforward if it is assumed that there is only one strong diffracted beam excited. Moreover, since this can be achieved quite easily experimentally, by orienting the crystal such that strong diffraction occurs from only one set of crystal planes, virtually all TEM is carried out with a two-beam condition:

a direct and a diffracted beam. When the direct, or transmitted, beam only is allowed to contribute to the final image by inserting a small aperture in the

Figure 5.27 Mechanism of diffraction contrast: the planes back focal plane to block the strongly diffracted ray,

to the RHS of the dislocation are bent so that they closely then contrast is shown on a bright background and is

approach the Bragg condition and the intensity of the direct known as bright-field imaging. If the diffracted ray

beam emerging from the crystal is therefore reduced . only is allowed through the aperture by tilting the

incident beam then contrast on a dark background is strong diffraction arises from near the defect. These observed and is known as dark-field imaging. These

diffracted rays are blocked by the objective aperture two arrangements are shown in Figure 5.26.

and prevented from contributing to the final image.

A dislocation can be seen in the electron microscope The dislocation therefore appears as a dark line (where because it locally changes the orientation of the crystal,

electrons have been removed) on a bright background thereby altering the diffracted intensity. This is illus-

in the bright-field picture.

trated in Figure 5.27. Any region of a grain or crystal The success of transmission electron microscopy

(TEM) is due, to a great extent, to the fact that it is is not strongly diffracting electrons. However, in the

possible to define the diffraction conditions which give vicinity of the dislocation the lattice planes are tilted

rise to the dislocation contrast by obtaining a diffrac- such that locally the Bragg law is satisfied and then

tion pattern from the same small volume of crystal (as small as 1 µ m diameter) as that from which the elec- tron micrograph is taken. Thus, it is possible to obtain the crystallographic and associated diffraction infor- mation necessary to interpret electron micrographs. To obtain a selected area diffraction pattern (SAD) an aperture is inserted in the plane of the first image so that only that part of the specimen which is imaged within the aperture can contribute to the diffraction pat- tern. The power of the diffraction lens is then reduced so that the back focal plane of the objective is imaged, and then the diffraction pattern, which is focused in this plane, can be seen after the objective aperture is removed.

The usual type of transmission electron diffraction pattern from a single crystal region is a cross-grating Figure 5.26 Schematic diagram illustrating (a) bright-field

pattern of the form shown in Figure 5.28. The simple and (b) dark-field image formation .

explanation of the pattern can be given by considering

148 Modern Physical Metallurgy and Materials Engineering

Figure 5.28 fcc cross-grating patterns (a) [0 0 1 ], (b) [1 0 1 ] and (c) [1 1 1 ].

the reciprocal lattice and reflecting sphere construc- tion commonly used in X-ray diffraction. In electron diffraction, the electron wavelength is extremely short

Ewald reflecting sphere is about 2.5 nm , which is about 50 times greater than g, the reciprocal lattice

are also small (about 10 radian or 1 2 ° for low-order

reflections) and hence the reflection sphere may be considered as almost planar in this vicinity. If the elec- tron beam is closely parallel to a prominent zone axis of the crystal then several reciprocal points (somewhat extended because of the limited thickness of the foil) will intersect the reflecting sphere, and a projection of the prominent zone in the reciprocal lattice is obtained,

i.e. the SAD pattern is really a photograph of a recip- rocal lattice section. Figure 5.28 shows some standard cross-grating for fcc crystals. Because the Bragg angle

for reflection is small ⊲³ 1 2 ° ⊳ only those lattice planes

which are almost vertical, i.e. almost parallel to the direction of the incident electron beam, are capable of Bragg-diffracting the electrons out of the objective aperture and giving rise to image contrast. Moreover, because the foil is buckled or purposely tilted, only one family of the various sets of approximately ver-

Figure 5.29 Schematic diagram to illustrate the tical lattice planes will diffract strongly and the SAD

determination of s at the symmetry position, together with pattern will then show only the direct beam spot and

associated diffraction pattern .

one strongly diffracted spot (see insert Figure 5.40). The indices g of the crystal planes ⊲hkl⊳ which are set at the Bragg angle can be obtained from the SAD.

contains electrons scattered through a smaller angle at Often the planes are near to, but not exactly at, the

P. Because P is a spherical source this rediffraction at Bragg angle and it is necessary to determine the precise

points such as Q and R gives rise to cones of rays deviation which is usually represented by the param-

which, when they intersect the film, approximate to eter s, as shown in the Ewald sphere construction in

straight lines.

Figure 5.29. The deviation parameter s is determined The selection of the diffracting conditions used from Kikuchi lines, observed in diffraction patterns

to image the crystal defects can be controlled using obtained from somewhat thicker areas of the specimen,

Kikuchi lines. Thus the planes ⊲hkl⊳ are at the Bragg which form a pair of bright and dark lines associated

angle when the corresponding pair of Kikuchi lines with each reflection, spaced jgj apart.

passes through 0 0 0 and g hkl , i.e. s D 0. Tilting of the The Kikuchi lines arise from inelastically-scattered

specimen so that this condition is maintained (which rays, originating at some point P in the specimen (see

can be done quite simply, using modern double-tilt Figure 5.30), being subsequently Bragg-diffracted.

specimen stages) enables the operator to select a spec- Thus, for the set of planes in Figure 5.30a, those

imen orientation with a close approximation to two- electrons travelling in the directions PQ and PR will

beam conditions. Tilting the specimen to a particular

be Bragg-diffracted at Q and R and give rise to rays orientation, i.e. electron beam direction, can also be

selected using the Kikuchi lines as a ‘navigational’ in the beam RR 0 originate from the scattered ray

in the directions QQ 0 and RR 0 . Since the electrons

aid. The series of Kikuchi lines make up a Kikuchi

PR, this beam will be less intense than QQ 0 , which

map, as shown in Figure 5.30b, which can be used to

The characterization of materials 149

2. High-angle information in the form of fine lines (somewhat like Kikuchi lines) which are visible in the direct beam and in the higher-order Laue zones (HOLZ). These HOLZ are visible in a pattern covering a large enough angle in reciprocal space. The fine line pattern can be used to measure the

lattice parameter to 1 in 10 4 . Figure 5.31b shows an example of HOLZ lines for a silicon crystal centred [1 1 1]. Pairing a dark line through the zero- order disc with its corresponding bright line through the higher-order disc allows the lattice parameter to

be determined, the distance between the pair being sensitive to the temperature, etc.

3. Detailed structure both within the direct beam and within the diffracted beams which show certain well-defined symmetries when the diffraction pat- tern is taken precisely along an important zone axis. The patterns can therefore be used to give crystal structure information, particularly the point group and space group. This information, together with the chemical composition from EELS or EDX, and the size of the unit cell from the indexed diffraction patterns can be used to define the specific crys- tal structure, i.e. the atomic positions. Figure 5.31c indicates the threefold symmetry in a CBDP from silicon taken along the [1 1 1] axis.

5.4.4.3 Higher-voltage electron microscopy The most serious limitation of conventional transmis-

sion electron microscopes (CTEM) is the limited thick- ness of specimens examined (50–500 nm). This makes preparation of samples from heavy elements difficult, gives limited containment of particles and other struc- tural features within the specimen, and restricts the study of dynamical processes such as deformation, annealing, etc., within the microscope. However, the usable specimen thickness is a function of the acceler- ating voltage and can be increased by the use of higher voltages. Because of this, higher-voltage microscopes

Figure 5.30 Kikuchi lines. (a) Formation of and (b) from (HVEM) have been developed. fcc crystal forming a Kikuchi map .

tilt from one pole to another (as one would use an Underground map).

1000 kV are only about one third of their correspond- ing values at 100 kV. One consequence of this is that

5.4.4.2 Convergent beam diffraction patterns an additional projector lens is usually included in high- When a selected area diffraction pattern is taken with

voltage microscope. This is often called the diffraction

a convergent beam of electrons, the resultant pattern lens and its purpose is to increase the diffraction cam- contains additional structural information. A ray dia-

era length so that the diffraction spots are more widely gram illustrating the formation of a convergent beam

spaced on the photographic plate. diffraction pattern (CBDP) is shown in Figure 5.31a.

The principal advantages of HVEM are: (1) an The discs of intensity which are formed in the back

increase in usable foil thickness and (2) a reduced ion- focal plane contain information which is of three types:

ization damage rate in ionic, polymer and biological specimens. The range of materials is therefore widened

1. Fringes within discs formed by strongly diffracted and includes (1) materials which are difficult to pre- beams. If the crystal is tilted to 2-beam conditions,

pare as thin foils, such as tungsten and uranium and these fringes can be used to determine the specimen

(2) materials in which the defect being studied is too thickness very accurately.

large to be conveniently included within a 100 kV

150 Modern Physical Metallurgy and Materials Engineering

Figure 5.31 (a) Schematic formation of convergent beam diffraction pattern in the backfocal plane of the objective lens, (b) and (c) h1 1 1 i CBDPs from Si; (b) zero layer and HOLZ (Higher Order Laue Zones) in direct beam and (c) zero layer C FOLZ (First Order Laue Zones) .

specimen; these include large voids, precipitates and some dislocation structures such as grain boundaries.

Many processes such as recrystallization, defor- mation, recovery, martensitic transformation, etc. are dominated by the effects of the specimen surfaces in thin samples and the use of thicker foils enables these phenomena to be studied as they occur in bulk mate- rials. With thicker foils it is possible to construct intri- cate stages which enable the specimen to be cooled, heated, strained and exposed to various chemical envi- ronments while it is being looked through.

A disadvantage of HVEM is that as the beam voltage is raised the energy transferred to the atom by the fast electron increases until it becomes sufficient to eject the atom from its site. The amount of energy transferred from one particle to another in a collision depends on the ratio of the two masses (see Chapter 4). Because the electron is very light compared with an atom, the transfer of energy is very inefficient and the electron needs to have several hundred keV before it can transmit the 25 eV or so necessary to displace an atom. To avoid radiation damage it is necessary to keep the beam voltage below the critical displacement value which is ³100 kV for Mg and ³1300 kV for Au. There is, however, much basic scientific interest in radiation damage for technological reasons and a HVEM enables the damage processes to be studied directly.