Itransm'"ed\
5.5 Observation of defects .1 Etch pitting
154 Modern Physical Metallurgy and Materials Engineering exciting a plasmon loss, Pi, to not exciting a plasmon, P0, is given by Pl/P0 = t/L, where t is the thickness, L the mean free path for plasmon excitation, and P i and P0 are given by the relative intensities of the zero loss and the first plasmon peak. If the second plasmon peak is a significant fraction of the first peak this indicates that the specimen will be too thick for accurate microanalysis.
The third region is made up of a continuous back- ground on which the characteristic ionization losses are superimposed. Qualitative elemental analysis can be carried out simply by measuring the energy of the edges and comparing them with tabulated energies.
The actual shape of the edge can also help to define the chemical state of the element. Quantitative analysis requires the measurement of the ratios of the intensities of the electrons from elements A and B which have suffered ionization losses. In principle, this allows the ratio of the number of A atoms, NA, and B atoms, NB, to be obtained simply from the appropriate ionization cross-sections, QK. Thus the number of A atoms will be given by
NA = (1/Q'~)[l~/lo]
and the number of B atoms by a similar expression, so that
B B A
NA/NB -- I ~ Q K / I K Q K
where I~ is the measured intensity of the K edge for element A, similarly for I~ and I0 is the measured intensity of the zero loss peak. This expression is similar to the thin foil EDX equation.
To obtain IK the background has to be removed so that only loss electrons remain. Because of the presence of other edges there is a maximum energy range over which IK can be measured which is about 5 0 - 1 0 0 eV. The value of QK must therefore be replaced by QK (A) which is a partial cross-section cal- culated for atomic transition within an energy range A of the ionization threshold. Furthermore, only the loss electrons arising from an angular range of scatter a at the specimen are collected by the spectrometer so that a double partial cross-section Q(A, ct) is appropriate.
Thus analysis of a binary alloy is carried out using the equation
NA Q~,(A, o~)I~(A, or)
NB Q'~(A, ot)l~(A, et)
Values of Q(A, c~) may be calculated from data in the literature for the specific value of ionization edge, A, oe and incident accelerating voltage, but give an analysis accurate to only about 5%; a greater accuracy might be possible if standards are used.
The characterization of materials 155
Figure 5.37 Direct observation of dislocations. (a) Pile-up in a zinc single crystal (after Gilman, 1956, p. 1000).
(b) Frank-Read source in silicon (after Dash, 1957; courtesy of John Wiley and Sons).
5.5.2 Dislocation decoration
It is well-known that there is a tendency for solute atoms to segregate to grain boundaries and, since these may be considered as made up of dislocations, it is clear that particular arrangements of dislocations and sub-boundaries can be revealed by preferential precip- itation. Most of the studies in metals have been carried out on aluminium-copper alloys, to reveal the dislo- cations at the surface, but recently several decoration techniques have been devised to reveal internal struc- tures. The original experiments were made by Hedges and Mitchell in which they made visible the disloca- tions in AgBr crystals with photographic silver. After a critical annealing treatment and exposure to light, the colloidal silver separates along dislocation lines. The technique has since been extended to other halides, and to silicon where the decoration is produced by diffus- ing copper into the crystal at 900~ so that on cooling the crystal to room temperature, the copper precipi- tates. When the silicon crystal is examined optically, using infrared illumination, the dislocation-free areas transmit the infrared radiation, but the dislocations dec- orated with copper are opaque. A fine example of dislocations observed using this technique is shown in Figure 5.37b.
The technique of dislocation decoration has the advantage of revealing internal dislocation networks but, when used to study the effect of cold-work on the dislocation arrangement, suffers the disadvantage of requiring some high-temperature heat-treatment dur- ing which the dislocation configuration may become modified.
5.5.3 D i s l o c a t i o n s t r a i n contrast in T E M The most notable advance in the direct observation of dislocations in materials has been made by the applica- tion of transmission techniques to thin specimens. The technique has been used widely because the disloca- tion arrangements inside the specimen can be studied.
It is possible, therefore, to investigate the effects of plastic deformation, irradiation, heat-treatment, etc. on the dislocation distribution and to record the move- ment of dislocations by taking cine-films of the images on the fluorescent screen of the electron microscope.
One disadvantage of the technique is that the materi- als have to be thinned before examination and, because the surface-to-volume ratio of the resultant specimen is high, it is possible that some rearrangement of dis- locations may occur.
A theory of image contrast has been developed which agrees well with experimental observations. The
156 Modern Physical Metallurgy and Materials Engineering basic idea is that the presence of a defect in the lattice causes displacements of the atoms from their position in the perfect crystal and these lead to phase changes in the electron waves scattered by the atoms so that the amplitude diffracted by a crystal is altered. The image seen in the microscope represents the electron intensity distribution at the lower surface of the speci- men. This intensity distribution has been calculated by a dynamical theory (see Section 5.5.7) which considers the coupling between the diffracted and direct beams but it is possible to obtain an explanation of many observed contrast effects using a simpler (kinematical) theory in which the interactions between the transmit- ted and scattered waves are neglected. Thus if an elec- tron wave, represented by the function exp (2rriko.r) where k0 is the wave vector of magnitude 1/Z, is inci- dent on an atom at position r there will be an elastically scattered wave exp (2rrikl.r) with a phase difference equal to 2rrr(kl - ko) when kl is the wave vector of the diffracted wave. If the crystal is not oriented exactly at the Bragg angle the reciprocal lattice point will be either inside or outside the reflecting sphere and the phase difference is then 2rrr(g + s) where g is the reciprocal lattice vector of the lattice plane giving rise to reflection and s is the vector indicating the devia- tion of the reciprocal lattice point from the reflection sphere (see Figure 5.39). To obtain the total scattered amplitude from a crystal it is necessary to sum all the scattered amplitudes from all the atoms in the crys- tal, i.e. take account of all the different path lengths for rays scattered by different atoms. Since most of the intensity is concentrated near the reciprocal lattice point it is only necessary to calculate the amplitude diffracted by a column of crystal in the direction of the diffracted by a column of crystal in the direction of the diffracted beam and not the whole crystal, as shown in Figure 5.38. The amplitude of the diffracted
beam
(~g
for an incident amplitude 4'0 = 1, is then/o'
4~g = (rri/~g) exp [-2Jri(g + s).rldr
and since r.s is small and g . r is an integer this reduces to
/o'
4~g = (~ri/~g) exp [-2n'is.rldr
/o'
= (Jri/~g) exp [-2~riszldz
where z is taken along the column. The intensity from such a column is
I~bgl 2 -- Ig -- [ n 2 / ~ ] ( s i n 2 7rts/(:rrs) 2)
from which it is evident that the diffracted intensity oscillates with depth z in the crystal with a periodicity equal to 1/s. The maximum wavelength of this oscil- lation is known as the extinction ~ distance ~g since the diffracted intensity is essentially extinguished at such positions in the crystal. This sinusoidal variation of intensity gives rise to fringes in the electron-optical image of boundaries and defects inclined to the foil surface, e.g. a stacking fault on an inclined plane is generally visible on an electron micrograph as a set of parallel fringes running parallel to the intersec- tion of the fault plane with the plane of the foil (see Figure 5.43).
In an imperfect crystal, atoms are displaced from their true lattice positions. Consequently, if an atom at r, is displaced by a vector R, the amplitude of the wave diffracted by the atom is multiplied by I$g = zrV cosO/~.F where V is the volume of the unit cell, 0 the Bragg angle and F the structure factor.
(a) incident beam (b)
Oo
Direct Diffracted wave wave ,,
z=O --ZStacking fault
~_ (~ _ _/ wedge crystal or
t
r(~ --/TJ" Extinction
,~ 7 / ' distance
\
, , z = t - . . .
Figure 5.38 (a) Column approximation used to calculate the amplitude of the d[ffracted beam qhg on the bottom surface of the crystal. The dislocation is at a depth y and a distance x from the column. (b) Variation of intensity with depth in a crystal.
The characterization of materials 157 an additional phase factor exp [2zri(kl - k0).R]. Then,
since (kl -- k0) = g + s the resultant amplitude is
f0 t
t~g =
(:rri/~g) exp [--2zri(g + s).(r + R)]dr If we neglect s.R which is small in comparison with g.R, and g.r which gives an integer, then in terms of the column approximation~0 t
t#g "-" (Tri/~g) exp (--2zrisz) exp ( - 2 n ' i g . R ) d z The amplitude, and hence the intensity, therefore may differ from that scattered by a perfect crystal, depending on whether the phase factor c~ = 2rrg.R is finite or not, and image contrast is obtained when
g.R :fi O.
5.5.4 Contrast from crystals
In general, crystals observed in the microscope appear light because of the good transmission of electrons. In detail, however, the foils are usually slightly buckled so that the orientation of the crystal relative to the electron beam varies from place to place, and if one part of the crystal is oriented at the Bragg angle, strong diffraction occurs. Such a local area of the crystal then appears dark under bright-field illuminations, and is known as a bend or extinction contour. If the specimen is tilted while under observation, the angular conditions for strong Bragg diffraction are altered, and the extinction contours, which appear as thick dark bands, can be made to move across the specimen. To interpret micrographs correctly, it is essential to know the correct sense of both g and s. The g-vector is the line joining the origin of the diffraction pattern to the strong diffraction spot and it is essential that its sense is correct with respect to the micrograph, i.e. to allow for any image inversion or rotation by the electron optics.
The sign of s can be determined from the position of the Kikuchi lines with respect to the diffraction spots, as discussed in Section 5.4.1.
5.5.5 Imaging of dislocations
Image contrast from imperfections arises from the additional phase factor c~ = 2rrg.R in the equation for the diffraction of electrons by crystals. In the case of dislocations the displacement vector R is essentially equal to b, the Burgers vector of the dislocation, since atoms near the core of the dislocation are displaced parallel to b. In physical terms, it is easily seen that if a crystal, oriented off the Bragg condition, i.e. s r 0, contains a dislocation then on one side of the dislocation core the lattice planes are tilted into the reflecting position, and on the other side of the dislocation the crystal is tilted away from the reflecting position. On the side of the dislocation in the reflecting position the transmitted intensity, i.e. passing through the objective aperture, will be
less and hence the dislocation will appear as a line in dark contrast. It follows that the image of the dislocation will lie slightly to one or other side of the dislocation core, depending on the sign of (g.b)s.
This is shown in Figure 5.39 for the case where the crystal is oriented in such a way that the incident beam makes an angle greater than the Bragg angle with the reflecting planes, i.e. s > 0. The image occurs on that side of the dislocation where the lattice rotation brings the crystal into the Bragg position, i.e. rotates the reciprocal lattice point onto the reflection sphere.
Clearly, if the diffracting conditions change, i.e. g or s change sign, then the image will be displaced to the other side of the dislocation core.
The phase angle introduced by a lattice defect is zero when g.R = 0, and hence there is no contrast, i.e.
the defect is invisible when this condition is satisfied.
Since the scalar product g.R is equal to g R cos 0, where 0 is the angle between g and R, then g.R --- 0 when the displacement vector R is normal to g, i.e. parallel to the reflecting plane producing the image. If we think of the lattice planes which reflect the electrons as mirrors, it is easy to understand that no contrast results when g.R -- 0, because the displacement vector R merely moves the reflecting planes parallel to themselves without altering the intensity scattered from them. Only displacements which have a component perpendicular to the reflecting plane, i.e. tilting the planes, will produce contrast.
A screw dislocation only produces atomic displace- ments in the direction of its Burgers vector, and hence because R = b such a dislocation will be completely 'invisible' when b lies in the reflecting plane producing the image. A pure edge dislocation, however, pro- duces some minor atomic displacements perpendicular to b, as discussed in Chapter 4, and the displacements give rise to a slight curvature of the lattice planes. An edge dislocation is therefore not completely invisible when b lies in the reflecting planes, but usually shows some evidence of faint residual contrast. In general,
To
,,, I )P
Figure 5.39 Schematic diagram showing the dependence of the dislocation image position on diffraction conditions.
158 Modem Physical Metallurgy and Materials Engineering
Figure 5.40
(a) Application of the g.b = 0 criterion. The effect of changing the diffraction condition (see diffraction pattern inserts) makes the long helical dislocation B in (a) disappear in (b) (after Hirsch, Howie and Whelan, 1960; courtesy of the Royal Society).however, a dislocation goes out of contrast when the reflecting plane operating contains its Burgers vector, and this fact is commonly used to determine the Burg- ers vector. To establish b uniquely, it is necessary to tilt the foil so that the dislocation disappears on at least two different reflections. The Burgers vector must then be parallel to the direction which is common to these two reflecting planes. The magnitude of b is usually the repeat distance in this direction.
The use of the g.b = 0 criterion is illustrated in Figure 5.40. The helices shown in this micrograph have formed by the condensation of vacancies on to screw dislocations having their Burgers vector b par- allel to the axis of the helix. Comparison of the two pictures in ( a ) a n d ( b ) s h o w s that the effect of tilt- ing the specimen, and hence changing the reflecting plane, is to make the long helix B in (a) disappear in (b). In detail, the foil has a [00 1] orientation and the long screws lying in this plane are 1/21110]
and 1/21110]. In Figure 5.40a the insert shows the 0 2 0 reflection is operating and so g.b =1/= 0 for either A or B, but in Figure 5.40b the insert shows that the 2 2 0 reflection is operating and the dislocation B is invisible since its Burgers vector b is normal
-- i
to the g-vector, i . e . g . b = 2 2 0 . 1 / 2 [ 1 1 0] = (~ • 1 • 2) + (~ x 1 • 2) + 0 = 0 for the dislocation B, and is 1
therefore invisible.
5.5.6 Imaging of stacking faults
Contrast at a stacking fault arises because such a defect displaces the reflecting planes relative to each other, above and below the fault plane, as illustrated in
Figure 5.41a. In general, the contrast from a stacking fault will not be uniformly bright or dark as would be the case if it were parallel to the foil surface, but in the form of interference fringes running par- allel to the intersection of the foil surface with the plane containing the fault. These appear because the diffracted intensity oscillates with depth in the crystal as discussed. The stacking fault displacement vector R, defined as the shear parallel to the fault of the por- tion of crystal below the fault relative to that above the fault which is as fixed, gives rise to a phase difference c~ = 2rrg.R in the electron waves diffracted from either side of the fault. It then follows that stacking-fault con- trast is absent with reflections for which c~ -- 2rr, i.e.
for which g.R = n. This is equivalent to the g.b = 0 criterion for dislocations and can be used to deduce R.
The invisibility of stacking fault contrast when g.R = 0 is exactly analogous to that of a dislocation when g.b -- 0, namely that the displacement vector is parallel to the reflecting planes. The invisibility when g.R = 1, 2, 3 . . . . occurs because in these cases the vector R moves the imaging reflecting planes normal to themselves by a distance equal to a multiple of the spacing between the planes. From Figure 5.41b it can be seen that for this condition the reflecting planes are once again in register on either side of the fault, and, as a consequence, there is no interference between waves from the crystal above and below the fault.
5.5.7 Application of dynamical theory
The kinematical theory, although very useful, has limi- tations. The equations are only valid when the crystal is
The characterization of materials 159
(a) Etectron beam
Top of foJt ; ; ....
1 I I ' 1 I
'l 'I i; I Ill I II '
i ~-vector
of: ' i reflecting
'I , ,I ,' : I,. ptanes : l J J I/
(b) .d, '1 ;
'-' I A, : : ~Reftecting
, , I, Y', i, : ', : ptanes displ.aced ,, I : i i/,_,, _, , , , 9 by spac,ng d
Stack,rig fault
Figure 5.41 Schematic diagram showing (a) displacement of reflecting planes by a stacking fault and (b) the condition forg.R = n when the fault would be invisible.
oriented far from the exact Bragg condition, i.e. when s is large. The theory is also only strictly applicable for foils whose thickness is less than about half an extinction distance ( ~ g ) and no account is taken of absorption. The dynamical theory has been developed to overcome these limitations.
The object of the dynamical theory is to take into account the interactions between the diffracted and transmitted waves. Again only two beams are consid- ered, i.e. the transmitted and one diffracted beam, and experimentally it is usual to orient the specimen in a double-tilt stage so that only one strong diffracted beam is excited. The electron wave function is then considered to be made up of two plane waves - a n inci- dent or transmitted wave and a reflected or diffracted wave
lp(r) = r -I- Cgexp(2~rikl.r)
The two waves can be considered to propagate together down a column through the crystal since the Bragg angle is small. Moreover, the amplitudes ~b0 and ~bg of the two waves are continually changing with depth z in the column because of the reflection of electrons from one wave to another. This is described by a pair of coupled first-order differential equations linking the wave amplitudes r and ~bg. Displacement of an atom R causes a phase change c~ --- 2rrg.R in the scattered wave, as before, and the two differential equations describing the dynamical equilibrium between incident and diffracted waves
d4~0 zri
m
dz
~g ~g
d ~ g z r i ( d R )
dE -- ~g t~0 "~ 27/'t~g S q- g-dz-z
These describe the change in reflected amplitude
t~g
because electrons are reflected from the transmitted wave (this change is proportional to 4~0, the transmitted wave amplitude, and contains the phase factor) and the reflection in the reverse direction.
These equations show that the effect of a displace- ment R is to modify s locally, by an amount propor- tional to the derivative of the displacement, i.e. dR/dz, which is the variation of displacement with depth z in the crystal. This was noted in the kinematical theory where d R / d z is equivalent to a local tilt of the lattice planes. The variation of the intensities 14~012 and Jt~gl 2
,.. 0,[
i^i '
---! _ ; : / Y - -
_ ! I k |
-3 -2 -1 0 1 2 3
t'/[g
Figure 5.42 Computed intensity profiles about the foil centre for a stacking fault with t~ = +2 zr/3. The full curve is the B.F. and the broken curve the D.F. image (from Hirsch, Howie et al., 1965).
for different positions of the column in the crystal, rel- ative to the defect, then gives the bright and dark-field images respectively. Figure 5.42 shows the bright- and dark-field intensity profiles from a stacking fault on an inclined plane, in full and broken lines, respectively.
A wide variety of defects have been computed, some of which are summarized below:
1. Dislocations In elastically isotropic crystals, perfect screw dislocations show no contrast if the condition g.b = 0 is satisfied. Similarly, an edge dislocation will be invisible if g . b - - 0 and if g.b x u = 0 where u is a unit vector along the dislocation line and b x u describes the secondary displacements associated with an edge dislocation normal to the dislocation line and b. The computations also show that for mixed dislocations and edge dislocations for which g.b • u < 0 . 6 4 the contrast produced will be so weak as to render the dislocation virtually invisible. At higher values of g.b x u some contrast is expected. In addition, when the crystal becomes significantly anisotropic residual contrast can be observed even for g.b = O.
The image of a dislocation lies to one side of the core, the side being determined by (g.b)s. Thus the image of a dislocation loop lies totally outside the core when (using the appropriate convention) (g.b)s is positive and inside when (g.b)s is neg- ative. "vacancy and interstitial loops can thus be distinguished by examining their size after chang- ing from + g to - g , since these loops differ only in the sign of b.