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III. Electron Diffraction
A. Principles of Diffraction
Diffraction is the spreading of waves around obstacles. One consequence of diffraction is that sharp shadows are not produced. The phenomenon is the result of interference and is most pronounced when the wavelength of the radiation is comparable to the linear dimensions of the obstacle. When a beam of light falls on the edge of an object, it will not continue in a straight line but will be slightly bent by the contact, causing a blur at the edge of the shadow of the object. The amount of bending will be proportional to the wavelength.
When a monochromatic beam of light falls on a single edge, a sequence of light and dark bands is produced; and with white light a sequence of colours much like the Newton colour sequence appears.
A diffraction grating consists of a regular two- or three-dimensional array of objects or openings that scatter light according to its wavelength over a wide range of angles. As these deflected waves interact, they reinforce one another in some directions to produce intense spectral colours. Diffraction arrays that reveal spectral colours in direct sunlight exist on the wings of some beetles and the skins of some snakes. Perhaps the most outstanding natural diffraction grating, however, is the gemstone opal. Electron microscope photographs reveal that an opal has a regular three-dimensional array of equal-size spheres, about 250 nm in diameter, which produce the diffraction.
Huygens showed that every point on a wave front may be regarded as a source of spherical wavelets, thus accounting for the laws of reflection and refraction. Fresnel added the hypothesis that the wavelets can interfere , and this led to a theory of diffraction.
- Interference Interference is the term used to describe the interaction of waves. These waves are typically electromagnetic radiation (visible light, x-rays, etc.) or matter waves (electrons, neutrons, etc.), although sound waves also behave this way. Any two (or more) waves whose wavefronts meet interfere with each other, that is, the resultant wave is the sum of the first two. Constructive interference occurs when the two initial waves are exactly in-phase - the peaks and troughs of one wave are aligned with the peaks and troughs of the other. Destructive interference occurs when the peaks of one wave coincide with the troughs of the other. When two waves of equal amplitude are exactly half a wavelength (π rad or 180°) out of phase, then their resultant wave has zero amplitude - they cancel out!
-1.
0
1
2 φ = 0
-1.
0
1
2
-1.
0
1
2 φ = 0
-1.
0
1
2 φ = 45°
φ
-1.
0
1
2 φ = 45°
-1.
0
1
2 φ = 45°
φ
-1.
0
1
2 φ = 90°
φ
-1.
0
1
2
-1.
0
1
2 φ = 90°
φ
-1.
0
1
2 φ = 135°
φ
-1.
0
1
2
-1.
0
1
2 φ = 135°
φ
-1.
0
1
2 φ = 180°
φ
-1.
0
1
2 φ = 180°
-1.
0
1
2
-1.
0
1
2
-1.
0
1
2 φ = 180°
φ
Interference also occurs between two wave trains moving in the same direction but having different wavelengths or frequencies. The resultant effect is a complex wave, and a pulsating frequency, called a beat , results when the wavelengths are slightly different.
- Fraunhofer ( L >> a ) Double Slit Diffraction This experiment is often attributed to Thomas Young , who first performed it in 1801 and thus helped prove the wave nature of light. A beam of light is first passed through a single slit as above, and subsequently passed through a double-slit system. The two slits have a width a and are separated by a distance d. The equation which describes the resultant intensity as a function of angle θ is:
θ λ
π θ β= λ
π β α=
α
α θ = cos where sin and sin
sin (^2) 2 a d I Im
The second factor on the right-hand side is the diffraction factor , while the last factor is the interference factor. The result looks similar to single-slit diffraction except the intensity is now modulated by the interference factor. The minima now occur at:
d sin θ = ( m +^12 )λ or ( ) ( )
( ) d
L m d m
Lm R^2
1 2 (^12) 2 2
(^12) ≈ λ +
λ − +
and the maxima at approximately :
d sin θ = m λ or d
L m d m
L m R
λ ≈ − λ
λ
2 2 2
where m = 0, 1, 2, 3... As with single-slit diffraction, it is possible to link the pattern on the screen to the microscopic nature of the slits. By accurately measuring the spacing of the new maxima or minima on the screen one can calculate the spacing of the slits, d. By measuring the distance between the unmodulated minima, it is also still possible to calculate the width of the slits, as before.
0
1
-20 -15 −10 -5 0 5 10 15 20
-20 -10 0 10 20
Relative Intensity
Distance (cm)
Angle (degrees)
L = 0.5 m
λ = 700 nm
a = 7 μ m
d = 15 μ m
d
λ L (^) = 2.3cm
single slit double slit, 15 μm apart
0
1
-20 -15 −10 -5 0
0
1
-20 -15 −10 -5 0 5 10 15 20
-20 -10 0 10 20
Relative Intensity
Distance (cm)
Angle (degrees)
L = 0.5 m
λ = 700 nm
a = 7 μ m
d = 15 μ m
d
λ L (^) = 2.3cm d
λ L (^) = 2.3cm d
λ L (^) = 2.3cm
single slit double slit, 15 μm apart
single slit double slit, 15 μm apart
- Fraunhofer ( L >> a ) Multiple Slit Diffraction The specific cases of single or double slit diffraction can be generalised for the case of an arbitrary number of slits, n. In this case, the equation of intensity becomes:
θ (^2)
2 2
2
sin
sin sin n I Im
The greater the number of slits, the sharper and more intense are the diffraction maxima. The same relationships between d and the maxima/minima spacings still hold and become more accurate the more slits that are included.
0
1
-20 -15 -10 -5 0 5 10 15 20
-20 -10 0 10 20
single slit six slits
Relative Intensity
Distance (cm)
Angle (degrees)
L = 0.5m
λ = 700 nm
a = 7^ μ m
d = 15^ μ m
n = 6 d
λ L (^) = 2.3cm
0
1
-20 -15 -10 -5 0 5 10 15 20
-20 -10 0 10 20
single slit six slits
Relative Intensity
Distance (cm)
Angle (degrees)
L = 0.5m
λ = 700 nm
a = 7^ μ m
d = 15^ μ m
n = 6
0
1
-20 -15 -10 -5 0 5 10 15 20
-20 -10 0 10 20
single slit six slits
Relative Intensity
Distance (cm)
Angle (degrees)
L = 0.5m
λ = 700 nm
a = 7^ μ m
d = 15^ μ m
n = 6 d
λ L (^) = 2.3cm d
λ L (^) = 2.3cm d
λ L (^) = 2.3cm
The above graphs have been normalised by a factor of 1/ n^2 for convenience. In reality, the intensity of the diffracted maxima are proportional to n^2 ; therefore, a diffraction grating consisting of thousands of slits can be used to yield very intense – and sharp – diffraction maxima.
- Fraunhofer ( L >> a ) Circular Aperture Diffraction When light from a point source passes through a small circular aperture, it does not produce a bright dot as an image, but rather a diffuse circular disc known as Airy's disc surrounded by much fainter concentric circular rings.
B. Quantum Electrodynamics (QED)
We have already mentioned the law of reflection, namely, that the angle of incidence equals the angle of reflection, θ i = θ f ; however, now it is useful to examine that law more closely. Consider the mirror shown in the figure below. The source, S, emits one photon at a time; and a detector is located at P. Let’s calculate the chance that a photon from S reaches the detector at P. In order to block the obvious straight path between S and P, we’ll put a screen at Q between them.
S P
Q
Light source
Light Detector
screen
Expected path Angle of incidence (^) θi of reflectionθrAngle of reflection mirror
S P
Q
Light source
Light Detector
screen
Expected path Angle of incidence (^) θi of reflectionθrAngle of reflection mirror
Now, we would expect that all the light that reaches P to have been reflected off the middle of the mirror, where the angle of incidence equals the angle of reflection, and that the far ends of the mirror have no role in the reflection. In fact, there are millions of routes for the photon to go from S to P, as shown below. Let’s look at these.
A B C D E F G H I J K L M
S P
Q
A B C D E F G H I J K L M
S P
Q
Each of these routes will take a different amount of time for the photon to reach P. The photon will clearly take longer to travel from, for example, S to A to P, than from S to G to P.†^ Because they’ve been travelling for different times, each photon will be slightly out of phase with its neighbours. If we represent the phase angle by an arrow (vector) whose direction corresponds to the relative phase of the photon (e.g., up = 0°, right = 90°, etc. ), then we can show graphically both the time required for each path and the relative phase of the photon upon reaching P.
† (^) Imagine yourself running from S to P – you’d hardly dash off to A first!
Time
A B C D E F G H I J K L M
Time
AA BB CC DD EE FF GG HH II JJ KK LL MM
You can think of the arrows as the direction of the hand of a stopwatch when the photon reaches P. By adding up all the arrows, we arrive at a resultant vector which represents the probability of a photon from S arriving at P.‡
It is apparent that the arrows from the ends of the mirror contribute very little to the overall resultant, and it is the middle section of the mirror which contributes most. This is true because it is in the middle section where the differences between adjacent paths are smallest, so the phase differences are also smallest. The ends of the mirror could easily be chopped off and the reflection would be virtually unaffected, but that does not mean that reflection is not happening there as well! We can test this theory by removing most of the mirror and leaving only a small section way out on the left. What’s left of the mirror is now in the wrong place for reflection, you’d think. We’ll divide this section up more finely now – fine enough so that there is not much difference in time between adjacent sections.
A OUT^ B OUT C
without scratches with scratches
A OUT^ B OUT C
without scratches with scratches
OUT OUT
without scratches with scratches
Now we see that some arrows point more or less to the right while others point more or less to the left. If we add all the arrows together, we again find they form more or less a circle and add up zero – no reflection. Now, if just the parts of the mirror where the arrows are pointing to the left are removed, the remaining arrows do indeed add up to a substantial resultant. Reflection does take place! Such a mirror is called a diffraction grating.
‡ (^) Actually, the probability is the square of the magnitude of this vector.
With this relation, it is quite straightforward to show that the wavelength of electrons accelerated by, say, 200 kV (as they are in the JEOL 2100) is 0.025 Å – much smaller than x-ray or typical neutron wavelengths. If such electrons are diffracted from the faces of the cubic crystal thallium chloride, for instance, the angle θ is only 0.186° and the total angular deflection of the electron beam is 0.37°. If a fluorescent screen or photographic film is placed a distance L from the crystal, the diffracted spot will be displaced from the undeviated beam by a distance R such that:
R = L tan 2 θ ≈ 2 θ L
since for very small θ, tan2θ ≈ 2 θ. Combining this relationship with Bragg’s law yields:
R L d d L R
λ λ
The factor L λ is often referred to as the camera constant and is generally calibrated experimentally for each microscope. This equation is used to index every kind of electron diffraction pattern.
Electron diffraction differs from x-ray diffraction in a number of ways. First , electrons are much less penetrating than x-rays. They are easily absorbed by air (where they can ionise air molecules) which means that the specimen and the recording/detecting device must all be enclosed in vacuum. Transmission electron diffraction patterns can only be obtained with specimens so thin as to be classed as foils (metals) or films. Second , electrons are scattered much more intensely than x-rays, so that even a very thin layer gives a strong diffraction pattern in a short time. Third , the intensity of electron diffraction decreases with increasing 2θ even more rapidly than for XRD, which means that the entire observable diffraction pattern is limited to an angular range of about ±4° 2θ. We will cover this topic in more detail later on.
The figure below shows the geometry of electron diffraction in the TEM. Since the wavelength of electrons is so small (0.025 Å at 200 kV), θ is very small (0.14° for d = 5 Å) and the Bragg equation reduces to 2d θ = λ. It is also apparent from the figure that tan2 θ = R/L , which reduces to 2 θ = R/L. Combining these two equations yields Rd = L λ, which is the equation used to index every electron diffraction pattern.
[001] electron diffraction pattern from an fcc crystal.
- The Structure Factor The resultant wave scattered by all the atoms in a unit cell is called the structure factor, F , because it describes how the atom arrangement affects the scattered beam. Mathematically, it is the sum of all the waves scattered by the individual atoms:
i ( hun kvn lwn ) n F (^) hkl fne
= ∑ 2 π 1
( )
where f is the atomic scattering factor (a function of sinθ/λ, tabulated in the International Tables for X-Ray Crystallography) and the summation extends over all the n atoms of the unit cell. The coordinates of the n th^ atom are ( unvnwn ) and those of the reflecting plane are ( hkl ). It is a complex number, combining both the amplitude and phase of the resultant wave. It can be re- written as:†
= (^) ∑ π + + + π + +
n F fn hun kvn lwn i hun kvn lwn 1
[cos 2 ( ) sin 2 ( )]
With this definition in mind, it is fairly easy to prove that:
( 1 ) where isan integer
where isanyinteger
2 4 6
3 5
e n
e e n
e e e
e e e
ni n
ni ni
i i i
i i i
π
π −π
π π π
π π π
† (^) Recall that eix (^) = cos x + i sin x
Geometry of electron diffraction in a TEM. L is the effective camera length, T is the transmitted spot, D is the diffracted spot, R is the distance between T and D , and 2 θ is the Bragg angle. The angle 2 θ is very small due to the small wavelength of the electrons, so Bragg’s Law reduces to 2d θ ≈ λ. Additionally, it can be seen that tan(2 θ ) = R/L , which reduces to 2 θ ≈ R/L. Combining these two equations yields Rd ≈ Lλ.
Sample (d = planar spacing)
T D
2 θ
R
L
2dsinθ = λ sinθ ≈ θ tan(2θ) = R/L tan(2θ) ≈ 2 θ ∴θ ≈ R/2L ∴ Rd ≈ Lλ
- Spot patterns The nature of the diffraction pattern is in indication of the number of grains contributing towards the pattern or the crystallinity of the specimen. A single crystal will give rise to a spot pattern, whereas diffraction from several crystals causes several overlapping spot patterns in which all the spots can be seen to lie along Debye rings. In the extreme case of hundreds of crystals contributing to the pattern, then the resulting spots are so closely spaced that they are no longer identifiable and merge into continuous rings (Debye rings). An amorphous sample results in no diffraction.
Single, perfect crystal
Diffraction pattern is regular array of dots
Small number of grains (crystals) with different orientation.
Diffraction pattern is discrete spots, each lying on a Debye ring
Large number of randomly oriented grains.
Diffraction pattern is continuous Debye rings
Amorphous material
No distinct diffraction pattern
- The reciprocal lattice The concept of the reciprocal lattice is one of convenience, as all spacings in a diffraction pattern are inversely proportional to the corresponding real spacing. It is constructed by converting each real-lattice plane ( hkl ) into a point at a distance g = 1/d hkl from the origin of the reciprocal lattice (d hkl is the real-lattice spacing of the ( hkl ) planes) along a direction perpendicular to ( hkl ). Some planes may not appear in the reciprocal lattice if they are forbidden by the extinction rules for the real-lattice crystal symmetry. If the real lattice has dimensions ao , bo , co , then the reciprocal lattice’s dimensions are a*^ = 1/ ao , b*^ = 1/ bo , c*^ = 1/ co ; and a*^ is normal to bo and co , b*^ is normal to ao and co , and c*^ is normal to ao and bo. The reciprocal lattice of an fcc crystal is a bcc one (and vice versa). Points in the reciprocal lattice are not “points” in the mathematical sense, but have some shape and size associated with them inversely proportional to the dimensions of the crystal or feature being examined (shape factor).
- The Ewald (or Reflecting) Sphere The Ewald sphere is another geometrical convenience for calculating when diffraction occurs for some arbitrary planes. A hypothetical sphere is centred on the point of electron scattering in the specimen and has a radius of 1/λ. Because the λ of electrons accelerated by, say, 100 kV, is only 0.037 Å, the radius of the sphere is 27 Å-1. The surface of this sphere can easily intersect more than one point in the reciprocal lattice, and so many spots can appear in a diffraction pattern. The size of the sphere is a major advantage of electron diffraction over, say, x-ray diffraction (XRD), which can excite only one Bragg reflection at a time. The Cu Kα radiation often used in XRD has λ = 1.54056 Å, with the Ewald sphere radius only 0.65 Å-1. Thus, the radius of the sphere is over 40 times bigger for 100 kV electrons, and the volume of the sphere is over 72, times bigger!
Using the Ewald sphere construction to predict which diffraction spots will occur is equivalent to the Bragg condition. The intersection of the sphere with successive lattice planes results in higher-order Laue zones (HOLZ). The zero-order Laue zone (ZOLZ) is typically used most often for indexing patterns, as most information is there and the first-order Laue zone (FOLZ) is often too far away to be imaged.
The reciprocal of an fcc real lattice is bcc.
000
(200) (220)
(020)
(111)
(202)
(002)
(222)
(022)
a (^) 1/a [110]
[010]
[100]
[111]
[001]
fcc bcc
a
111 111
220
220
111 111
b
111 111
220
220
111 111
002
002
442
442 006
006
(a) Kinematic simulation and (b) experimental DP of fcc Nd 2 Hf 2 O 7 with the beam parallel to [110] (zone axis = [110]).
- Kikuchi Lines Kikuchi lines, first observed by S. Kikuchi in 1928, are another consequence of the dynamical theory. When electrons are incident on a sample, some of the electrons are inelastically scattered (they lose energy). Since more scattered electrons lose small amounts of energy than large amounts, the wavefront of the incident beam is biased in the forward (downward) direction. Many electrons are scattered through an angle α and now satisfy the Bragg condition for an arbitrary set of planes.
These electrons end up at E. Fewer electrons scatter through the larger angle β and are subsequently diffracted by the same planes to D. Since more electrons are re-directed towards E than are lost in going to D, the result is a bright line at E and a dark line at D. T is the transmitted spot and g is the spot corresponding to the reflecting planes ( hkl ). When the sample is oriented exactly at the Bragg condition for ( hkl ), D passes through the centre of T and E passes through g. The distance between E and g is known as the deviation parameter and is a very accurate way of measuring the angular deviation from the exact Bragg condition.
E g D T
α
β
2 θ
2 θ
s > 0
sLd
s = 0 (Bragg)
s < 0
λL/d
g T
s > 0
sLd
s = 0 (Bragg)
s < 0
λL/d
s > 0
sLd
s = 0 (Bragg)
s < 0
λL/d
g T
In reality, the bright and dark lines observed on flat DP negatives are hyperbolic intersections of the film with cones (Kikuchi or Kossel cones), one bright and one dark. The excess line is always further away from T than the deficit line since α < β.
Kikuchi lines are very useful in orienting the crystal precisely in the TEM. When the specimen is tilted, the Kikuchi lines move as if they were attached to the bottom of the crystal. The various diffracted spots never move, they just appear and disappear as the angle changes; but Kikuchi lines move with the specimen. Following them is a handy way to tilt systematically in order to navigate from one zone axis to another or to find appropriate two-beam conditions (where only one set of planes Bragg diffracts) for defect analysis.
Kikuchi lines are only observable if the specimen is thick enough to generate sufficient intensity of scattered electrons. In a very thin specimen the lines will be too weak to distinguish from the background. As thickness increases, Kikuchi lines and then bands are observed until finally total absorption occurs and nothing is visible.
- Crystal Shape Factor Since all distances (lengths) in reciprocal space are 1/(real-space distances), the shape of the TEM specimen has an effect on the shape of the reciprocal lattice points. A disk-shaped TEM sample is very thin (about 100nm in the z direction) but has a comparatively large area ( x and y directions). Consequently, a disk-shaped specimen will produce electrons distributed along a rod
hkl ’s. With this rule, the spots are made systematic by indexing g 1 as (1 1 1 , ) g 2 as ( 220 , and) g 3 as (1 1 1. In this way, ) g 1 + g 3 = ( 1 1 1 +) (1 1 1 = ) ( 220 =) g 2. Every other spot in the pattern is now determined by similar vector addition.
111 111
220
220
111 111
002
002
442
442 006
006
g1 g g
Since the camera constant is typically only approximately known, the ratios of d -spacings should be checked to ensure that they match those calculated from the JCPDS card. Even if the absolute value of the d ’s is slightly off, the ratio of d ’s should be very nearly an exact match. As an additional check, the angles between each plane should be measured to ensure that they are correct. For cubic crystals, the angle φ between two planes (two plane normals), h 1 k 1 l 1 and h 2 k 2 l 2 , is given by:
cosφ =
h h k k l l
h k l h k l
1 2 1 2 1 2
1
2 1
2 1
2 2
2 2
2 2
2
- Zone Axis Identification If the planes in a diffraction pattern have been indexed correctly, then the zone axis [mnp] is defined as the direction which they all have in common. The g vectors of all the planes must be perpendicular to this direction, i.e. , [hkl] • [mnp] = 0. It is determined by finding two spots, h 1 k 1 l 1 and h 2 k 2 l 2 , which are not on the same systematic row. Then, m = k 1 l 2 - k 2 l 1 , n = l 1 h 2 - l 2 h 1 , p = h 1 k 2 - h 2 k 1. A simple way is to write down the plane indices in two columns and strike out the first and last digits in each column. Then, a simple cross-multiplication gives the correct result. Sometimes, a high-index direction will be calculated, e.g. , [220], as in this case. For crystallographic directions , one can legitimately multiply through by any number and still arrive at an equivalent direction. Multiplying [220] through by ½ yields [110], which is the lowest index possible for this direction and so is reported as the direction of the zone axis (the magnitude of vectors [220] and [110] are different, but their directions are crystallographically identical).
Diffraction pattern from fcc Nd 2 Hf 2 O 7.
Of course, this is exactly equivalent to taking the cross product of the two g vectors:
( 1 1 ) ( 1 1 ) ( 1 1 ) [ 220 ] [ 110 ]
det 1 1 1 = i + − j − − + k − + = =
i j k
The Weiss zone law says that for any plane ( hkl ) contributing to a DP along a given zone axis [ mnp ], then: m h + n k + p l = 0
More formally, this rule takes the form of a dot product:
∴ θ= ∴θ= °
× θ=
cos 0 90
cos 0
( ) [ ] 0
hkl mnp
hkl mnp
g r 0
[ mnp ] h1 k1 l1 h1 k1 l
h2 k2 l2 h2 k2 l
1+1,1+1,-1+1 = [ 220 ]= [110]
(a) Generic illustration of a zone axis. (b) Determination of the zone axis of the diffraction pattern in the figure above.
a b
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
