MSE 421/521 Structural Characterization
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 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 due to interference
; 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.
1. 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!
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φ = 0
0.5
-0.5
φ = 90°
-2
φ = 180°
φ
= 45°
φ = 135°
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.
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2. Fraunhofer ( L >> a ) Single Slit Diffraction
Plane waves that pass through a restricted opening emerge as divergent waves. When the opening is less than one wavelength in diameter the emergent wave is nearly spherical.
Whenever a beam of light is restricted by holes or slits or by opaque obstacles that block out part of the wave front, some spreading occurs at the edges of geometrical shadows.
I
θ
= I m

 sin α
α


2
where α =
π
λ a sin θ
The minima occur at sin θ = m λ a
or R = a 2 m
−
λ L m 2
λ
2
≈ m λ a
L where a is the slit width and m is an integer ( m = 1, 2, 3...) and R is the distance to the minimum as measured on the screen at Q . The approximation for R assumes L >> R .
What is projected onto the screen Q is a pattern similar to the one below. The width of the pattern is inversely proportional to the slit width
, and when a = λ , θ = 90°, which indicates that the central bright band fills the entire screen.
From the resulting interference pattern that emerges on the screen at Q, it is possible to calculate the width of the slit by accurately measuring the spacing of the minima on the screen. Thus, diffraction is a way of linking measurable macroscopic phenomena to microscopic ones.
λ = 700nm
µ m
λ a
L
= 5cm
0
-20
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3. 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:
I
θ
= I m


 sin
α
α 


2 cos 2
β where α =
π
λ a sin θ and β =
π
λ d sin θ
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 + 1
2
) λ or R = d 2
L
λ
2
( m + 1
2
)
−
( m + 1
2
) 2
≈
L λ ( m + 1
2
) d and the maxima at approximately : d sin θ = m λ or R = d 2
L λ
− m m 2
λ
2
≈
L λ d m 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.
λ = 700 nm
µ m
µ m
λ d
L single slit double slit,
15 µ m apart
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4. 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:
I
θ
= I m


 sin
α
2
2
α  



 sin 2 sin 2 n β
β



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.
λ = 700 nm
µ m
µ m
λ d
L intensity of the diffracted maxima are proportional to n 2
The above graphs have been normalised by a factor of 1/ n 2 for convenience. In reality, the
; therefore, a diffraction grating consisting of thousands of slits can be used to yield very intense – and sharp – diffraction maxima.
5. 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.
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λ = 700nm
µ m m λ L
For small angles θ
The intensity is given by:
I =
π
2
4 a 4 

J
1
(
π
π aR aR /
/
L
L λ )
λ


2 where J
1
( z ) is a Bessel function of the first kind, so the mimima occur when J
1
(
π aR / L λ
) = 0 (the zeros of the Bessel function of the first kind are 3.831706, 7.0155867, etc.
) and the maxima correspond to the maxima in the Bessel expression.
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C. X-Ray Diffraction (XRD)
Bragg’s Law
The same principles of multi-slit diffraction or diffraction gratings with visible light can be applied to finer gratings, i.e.
, crystals, with photons of shorter wavelengths, e.g.
, x-rays.
In deriving the equation which describes this kind of diffraction, W.L. Bragg made several assumptions. First, the incoming radiation is assumed to be parallel
. Second, the x-ray source must be monochromatic
(all photons of the same wavelength). There are various ways of producing such a monochromatic beam, but most involve the use of a filter and result in an approximately monochromatic beam. Thirdly, the crystal is assumed to be perfect
. Fourthly, only beams which are diffracted at an angle equal to the incident angle
are considered – one reason many people refer to x-ray diffraction peaks, somewhat incorrectly, as reflections. d sin θ d sin θ
When two x-ray photons are incident on a crystal, as shown above, the lower of the two must travel a greater distance (and so take longer than the other photon) to reach the detector. If the planar separation is d , then the extra path difference is d sin θ + d sin θ , or 2 d sin θ . When they start out from the source, both photons are in phase, but the time required for the lower photon to traverse this extra distance will necessarily change its phase relationship with the top photon.
If this extra distance is an integer, n , multiple of the wavelength of the x-ray, then the two waves are again in phase (or exactly 360 n ° out of phase, which comes to the same thing).
Mathematically, we have:
2 d sin θ = n λ which is called the
Bragg Law
. In general, it is difficult to satisfy, as strong diffraction will only occur for certain combinations of d , λ , and 2 θ . In order to increase the chances of it being satisfied for all the possible planes in a crystal, we have to either vary λ or 2 θ . By scanning a range of 2 θ along a great arc, more diffracted peaks will be obtained; however, for reasons to do with the intensities of these peaks, it is more common to use a polycrystalline powder than a single crystal in such an experiment. If a random orientational distribution can be maintained in the powder, then all possible diffracted peaks should be obtainable and an exact value for intensities calculated from the crystal structure and temperature. If the orientational distribution is not completely random, the sample is said to be textured
– and a single crystal is the most extreme form of preferred orientation or texturing.
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X-Ray Diffractometer
Operational principle
A diffractometer
works by shining a monochromatic beam of x-rays on a (usually) powder sample and a detector moves slowly from one angle to another, recording the intensity of diffracted x-rays as it moves. Newer diffractometers have stationary detectors which are capable of detecting photons at many angles at once, this speeding up the acquisition of patterns. The pattern of diffracted beams created in this way yields two kinds of information. The exact positions
of the peaks (2 θ ) contain information about the size of the unit cell ( d ). The intensity of the peaks gives information about the detailed atomic arrangement within the unit cell. In any case, the limitations of the Bragg law require that:
λ
= sin θ < 1 ∴ d >
λ
2 d 2
In order for a diffraction peak to appear, the condition that d > λ /2 must be satisfied. If d is too small for a given λ , it will never appear no matter what the angle; therefore, in order to observe very small values of d , small values of λ must also be used. An example of such a diffraction pattern recorded by a Siemens D5000 diffractometer is shown below.
r
λ =1.540562Å (Cu)
Diffraction Angle (2 θ )
Generation of X-Rays
At the heart of the diffractometer is the source of incident x-rays. In order to produce these x-ray photons, a metal target (typically Cu or Mo) is bombarded with high-energy electrons (typically
30 – 50 keV) in a vacuum. When the electrons slam into the target, their deceleration causes xrays to be emitted. In other words, their kinetic energy is transformed into photons of x-ray radiation. These x-ray emissions are sometimes called heterochromatic, polychromatic, continuous, or white radiation, as it is made up, like white light, of a mixture of rays with many
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MSE 421/521 Structural Characterization wavelengths. It is also called
Brehmsstrahlung
(German for “braking radiation”) because it is caused by electron deceleration. As we shall see later, these electrons are also capable of ejecting an inner electron from the atom, thus leaving a hole in the lower shell and putting the atom in an excited state. An outer electron can fill this hole by dropping down into it, releasing a quantum of energy via an x-ray photon. The energy of this photon will correspond exactly to the quantum energy difference between the outer and inner shells of the atom, which is a specific value for each species of atom. Wavelengths for the x-ray photons released by transitions of electrons from the L to K shell (so-called K α photons) and from the M to K shell (K β photons) in some elements are:
Element Wavelength (Å)
Cr
Fe
Co
Ni
Cu
Mo
K α K β
2.29100 2.08487
1.937355 1.75661
1.790260 1.62079
1.659189 1.500135
1.541838 1.392218
0.710730 0.632288
In reality, two finely separated wavelengths are observed for K α (K α
1
and K α
2
) for each element due to the exchange of angular momentum between the electron and the x-ray photon, and this subject will be dealt with more thoroughly later on.
Collimation and Filtering
The incident x-rays pass through a collimator
to produce a nearly parallel beam of x-rays, as required by the Bragg equation. The beam is then sometimes passed through a filter
, which works by allowing photons with wavelengths corresponding to K α to pass while absorbing photons of lower wavelengths (e.g., K β ). Such a material is said to have a K absorption edge between K α and K β .
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Schematic of copper radiation (a) before and (b) after passing through a nickel filter. The dashed line is the mass absorption coefficient for nickel.
This beam is then incident on the sample
. Specimens are normally in the form of powders, but single crystals or flat sheets can also be mounted. The specimen holder can be rotated to reduce texturing effects. Normally, samples are rotated through θ while the detector rotates through 2 θ to maintain the optical geometry. Diffracted beams are then sent towards the detector.
Sometimes a monochromator
is used to strip out extraneous radiation and leave just K α . A monochromator in a diffractometer can be placed either in the diffracted (secondary) or incident
(primary) beam. Placement of the monochromator in the diffracted beam has the advantage of suppressing background radiation originating in the specimen such as fluorescence and Compton scattering . Monochromators are often made from crystals of either NaCl, LiF, SiO
2
(quartz), Al,
Ge, or graphite. These crystals work by only diffracting the K α component of the x-ray beam, and this reflected beam is then used either as the incident (primary) beam or is sent to the detector in the diffracted (secondary) beam.
X-ray detection
There are several kinds of x-ray detectors. In a proportional counter
, the diffracted x-rays are detected by a simple gas ionisation process. X-rays are incident on the device and pass through an x-ray-transparent window into a chamber filled with an inert gas, e.g.
, Ar. The x-rays ionise gas atoms, and the free photoelectrons (which we will cover in more detail later) and Compton recoil electrons created accelerates towards the wire anode by a 1000 V potential difference. The positively charged gas ions move towards the casing. Because the field is so intense, the electrons acquire a great deal of energy on their race to the wire – so much so that they are able to knock additional electrons off atoms they hit along the way, thus causing a cascade of electrons or gas amplification . The current in the wire is proportional to the numbers of x-rays arriving, i.e.
, intensity. If the voltage is reduced to, say, 200 V, the result is what is usually called an ionisation chamber
. Freed electrons will no longer have sufficient energy to further ionise atoms and cause amplification. Such a device was used in the original Bragg spectrometer but is now obsolete because of its low sensitivity. If, on the other hand, the voltage is increased to, say, 1500 V, the device will act as a
Geiger counter
. Because they are unable to count at high rates, the Geiger counter is now also obsolete in diffractometry.
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≈ 1000V proportional counter
A scintillation counter
exploits the ability of some materials to fluoresce visible light. The amount of light emitted is proportional to the x-ray intensity and can be measured by means of a photomultiplier. Diffracted x-rays are incident on a crystal of sodium iodide (NaI) activated with a small amount of thallium. It fluoresces violet light when bombarded with x-rays. A photon of light is produced ( scintillation
) for every x-ray photon absorbed. This light photon passes into the photomultiplier, where it first encounters the photocathode material (typically CsSb). The photocathode emits one electron for every photon of light absorbed, and these electrons are drawn through a series of several dynodes , each maintained at a potential about 100 V more positive than the preceding one. On reaching the first dynode, each electron from the photocathode knocks out several more from the metallic dynode surface, each of which knock out several more from the next dynode, and so on. The result is an enormous final pulse about the same size as that in a Geiger counter. The whole process takes less than a microsecond, so a scintillation counter can operate at high count rates
(10 5 /s) without loss. Scintillation counters are also more sensitive to high-frequency hard radiation than proportional counters. The counter on the Siemens D5000 diffractometer in the IRC is a scintillation type. glass photocathode, CsSb ele ctro n n
Tl-NaI scintillation counter
Like the proportional counter, the pulse size is proportional to the energy of the photons absorbed; however, the pulse is much less sharply defined, making it more difficult to
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MSE 421/521 Structural Characterization discriminate between x-rays of different wavelengths (energies) on the basis of pulse size with a scintillation counter.
Semiconductor Counters
are the newest form of counter. They produce pulses proportional to the absorbed x-ray energy with better energy resolution than any other counter. Boron and lithium-doped silicon, denoted Si(Li) and inevitably called “silly”, is used for x-ray detection.
The surfaces of the 3-5 mm thick crystals are oppositely charged by the diffusion of Li high reverse bias, leaving most of the bulk with an equal concentration of B +3
+1 under a
and Li +1 and thereby intrinsically semiconducting
. The crystal can now function more or less like a solidstate ionisation chamber. When an x-ray is incident on the semiconductor, about 1000 electronhole pairs can be created. If a high voltage (1 kV) is maintained across the faces of the crystal, the electrons and holes will be swept to these faces, creating a pulse in the external circuit.
Because there is no charge amplification in the detector, a field-effect transistor (FET) is used to amplify the signal to the mV range.
A disadvantage of this type of counter is that it must be operated and maintained at cryogenic temperatures to minimise the thermal current (caused by thermally created electron-hole pairs) and diffusion of lithium. The FET should also be cooled during operation to maximise resolution, thus necessitating a bulky cryostat to hold several liters of liquid nitrogen.
During an actual experiment, the operator can normally choose the data collection time required by manipulating the angular speed of the detector (degrees/minute), the step size (typically 0.02° per step), and the dwell time at each angle (typically 1 – 4 seconds). For a diffractometer with a moving detector, tests can take a few minutes to several hours. Acquisition is much quicker for a parallel detector which can detect diffracted x-rays at many angles simultaneously.
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Intensity of Diffracted Rays
The relative intensity of diffracted beams in a diffractometer (unpolarised) is given by:
I = F 2
 p


1 + sin 2 cos
θ
2 2 cos
θ
θ

 e

− 2 M
We shall now look at each part of this equation separately in more detail.
The Structure Factor
In the above equation, F is the resultant wave scattered by all the atoms in a unit cell. It is called the structure factor 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:
F
( hkl )
= n
∑
1 f n e 2 π i ( hu n
+ kv n
+ lw n
) where f is the atomic scattering factor (a function of sin θ / 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 ( u n v n w n
) 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 rewritten as: †
F = n
∑ f n
[cos 2
π
( hu n
+ kv n
+ lw n
)
+ i sin
1
With this definition in mind, it is fairly easy to prove that:
2
π
( hu n e π i
= e 3 π i
= e 5 π i
= − 1
+ kv n
+ lw n
)] e 2 π i e n π i
= e 4 π i
= e 6 π i
= + 1
= e − n π i where n is any integer e n π i
= ( − 1 ) n where n is an integer
These simple rules enable us to calculate the structure factors for most reflections; however, there are certain special cases in which we are required to use the trigonometric functions instead, e.g.
, if ( hu n
+ kv n
+ lw n
)
≠
0, ½, 1...).
† Recall that e ix = cos x + i sin x
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Example Calculation One: Simple Cubic
The simplest case is that of a unit cell consisting of a single atom ( n = 1) at the origin. In this case,
F = fe 2 π i ( h 0 + k 0 + l 0 )
= fe 2 π i ( 0 )
= fe 0
= f ∴ F 2
= f 2
In this case, F 2 is independent of ( hkl ) and is the same for all reflections.
Example Calculation Two: BCC
In a body-centred unit cell (it need not be cubic) with one atom at the origin and an additional one in the body centre (½,½,½), the calculation is a bit longer...
F = fe 2 π i ( h 0 + k 0 + l 0 )
+ fe 2 π i ( 1
2 h +
1
2 k +
1
2 l )
= f + fe π i ( h + k + l )
= f ( 1 + e π i ( h + k + l ) )
So, if ( h + k + l ) is an even integer,
F = f ( 1 + 1 ) = 2 f but if ( h + k + l ) is an odd integer,
F = f ( 1 − 1 ) = 0
In this case, the intensity of the reflection is zero (no reflection will occur). Such reflections are said to be forbidden
and include, e.g.
, (100), (111), (210), etc .
Alternatively, if the two atoms in the unit cell are different elements (therefore with different atomic scattering factors, f
1
and f
2
), then
F = f
1 e 2 π i ( h 0 + k 0 + l 0 )
+ f
2 e 2 π i ( 1
2 h + 1
2 k + 1
2 l )
= f
1
+ f
2 e π i ( h + k + l ) and
F = f
1
+ f
2
if ( h+k+l ) is an even integer. These planes will be strongly diffracting.
F = f
1
− f
2
if ( h+k+l ) is an odd integer. These planes will be only weakly diffracting.
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Example Calculation Three: FCC
In a face-centred unit cell (it need not be cubic) with one atom at the origin (0,0,0,) and one in each of the face centres (½,½,0), (½,0,½), and (0, ½,½)
F = fe 2 π i ( h 0 + k 0 + l 0 )
+ fe 2 π i ( 1
2 h +
1
2 k + 0 l )
+ fe 2 π i ( 1
2 h + 0 k +
1
2 l )
+ fe 2 π i ( 0 h +
1
2 k +
1
2 l )
= f ( 1 + e π i ( h + k )
+ e π i ( h + l )
+ e π i ( k + l ) )
When h , k , and l are all odd or all even, the sum of any two of them will be even; hence, ( h + k ) ,
( h + l ) and
F = 4 f
( k + l ) are always even. In this case,
On the other hand, if h , k , and l are mixed odd and even, then the sum of the three exponentials is always -1. For example, if ( hkl ) = (120), then the exponentials are (-1 -1 + 1) = -1. If ( hkl ) =
(130), then (1 -1 -1) = -1. In this case,
F = f ( 1 − 1 ) = 0
Clearly, diffraction maxima from planes of mixed index are forbidden in this case.
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Example Calculation Four: NaCl
Now let’s look at a real crystal – the simple case of NaCl. This crystal has face-centred symmetry and, because it consists of two different ions, can be described as two interpenetrating fcc lattices offset from each other by (½ 0 0); therefore, there are eight ions per unit cell. The coordinates are:
Na: (0 0 0) (½ ½ 0) (½ 0 ½) (0 ½ ½)
Cl: (½ ½ ½) (½ 0 0) (0 ½ 0) 0 0 ½)
F
F
=
= f
Na e 2 π i f
Na
(
1 +
+ e f
Na e
2
π i
( h
2
+ k
2
)
π i ( h + k )
+ e
+ f
Na
π i ( h + l ) e
2
+
π i e
( h
2
+ l
2
)
π i ( k + l )
+
) f
+
Na f e
2
π i
Cl
(
( k
2
+ l
2
) e π i ( h
+
+ k + l f
Cl e
2
π i
( h
2
+ k
2
)
+ e π ih
+
+ l
We can factor out an e  i ( h+k+l ) from the Cl term and obtain:
F = f
Na
( 1
+ e π i ( h + k )
+ e π i ( h + l )
+ e π i ( k + l ) )
+ f
Cl e π i ( h + k + l ) ( 1
+ e − π i ( k + l )
2
)
+ e π ik
+ f
Cl e
2
π i
+ e π il h e − π i ( h + l )
)
+
+ f
Cl e
2
π i k e − π i ( h + k )
+
) f
Cl e
2
π i l
Now, because e n
 i = e n
 i , the signs of the exponents in the Cl term can be changed:
F = f
Na
(
1 + e π i ( h + k )
+ e π i ( h + l )
+ e π i ( k + l ) )
+ f
Cl e π i ( h + k + l ) (
1 + e π i ( k + l )
+ e π i ( h + l )
+ e π i ( h + k ) )
This expression can be re-grouped:
F =
(
1 + e π i ( h + k )
+ e π i ( h + l )
+ e π i ( k + l ) )( f
Na
+ f
Cl e π i ( h + k + l ) )
Here, the first set of parentheses corresponds to the face centring, whereas the second corresponds to the basis of the unit cell – the one Na + and one Cl associated with each lattice point. The first part vanishes if h , k , and l are mixed odd and even, otherwise it is equal to 4. For unmixed indices (all odd or all even),
F = 4
( f
Na
+ f
Cl e π i ( h + k + l ) )
= 4 ( f
Na
+ f
Cl
) if ( h + k + l ) is even
= 4 ( f
Na
− f
Cl
) if ( h + k + l ) is odd
Introducing a second atom type has not eliminated any reflections allowed, but it has decreased the intensity of some. For example, the (111) reflection now involves the difference, rather than the sum, of the scattering powers of the two ions. This result is independent of which way round the Cl and Na + are labelled in the unit cell. It is the relative position of the atoms which is important.
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Extinctions due to glide
Say we have c glide plane perpendicular to b (recall c glide involves reflection across a plane whose normal is perpendicular to c and a half translation along c ).
For every atom at xyz we generate another one at x ,y , z +½.
The structure factor is:
π i(hx + ky + exp
[
2 π i(hx ky + l ( z + ½ ) ] } F hkl
= f { exp
For the special case of k = 0:
[
2 lz)
]
+
F h0l
= f
{ exp
[
2 π i(hx + lz)
]
+ exp
[
2 π i(hx + l
( z +
½ ) ] }
= f
= f


 exp
[
2 π i(hx
{ exp
[
2 π i(hx +
+ lz)
]


1 + exp


2 π il
2 lz)
] [
1 + ( − 1) l
] }







= 0 if l = 2 n + 1
= 2 fexp
[
2 π i(hx + lz)
] if l = 2 n
Reflections of type { h 0 l } will therefore be missing if l is odd.
{ h 0 l }: l = 2 n
Extinctions due to screw axes
Consider a 2
1
screw axis parallel to b .
The symmetry-related positions are xyz , x , y +½,z .
F
0k0
= f
{ exp
[
2 π i(ky)
]
+ exp
[
2 π i(k(y + ½) ] }
=
= f


 exp
[
2 π iky
]


1 + f
{ exp
[
2 π iky)
] [
1 + exp


( − 1) k
] }
2 π ik
2







= 0 if k = 2 n + 1
= 2 fexp
[
2 π iky
] if
Thus reflection condition: {0k0}: k = 2 n k = 2 n
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MSE 421/521 Structural Characterization
Extinctions due to centring
Centring Type Condition
P None
I
F
A
B h h k h
+
+
+
+ l l k k
+ l = 2
= 2
= 2
= 2 n n n , n k + l = 2 n , h + l = 2 n
C
R
Phase Problem
h h
+
+ k k
= 2
+ l n
= 3 n (hexagonal axes)
Intensities of diffraction effects, I hkl
α | F hkl
| 2
The electron density function requires knowledge of F , whereas diffraction intensities only give
F 2 .
But F is a complex number, that is, F = A + iB and F 2 = ( A + iB )( A – iB ) = A 2 + B 2
So intensities only give us ( A 2 + B 2 ) and not A and B directly.
E.g., if we know that F 2 = 10, then we also know that A 2 + B 2 = 10, but there are an infinite number of combinations of A and B which satisfy this condition:
(±1, ±3), (±3, ±1), (±2, ± 6 ), ( ± 6 ,±2), ( ± 5 , ± 5 ), ( ± 10 ,0), (0, ± 10 )...
In order to calculate the crystal structure (atomic coordinates) from the electron density function, we need to know F , which means we need to know A and B independently . But diffraction only gives us F 2 and so ( A 2 + B 2 ) = phase problem !
Luckily there are methods we can use to get around this problem.
Effect of thermal vibration
Atoms undergo thermal vibrations about their mean positions even at absolute zero, and the amplitude of these vibrations increases with temperature. Increased vibrations has three main effects:
1.
It expands the unit cell, changing all the d -spacings and 2 θ positions of diffraction peaks.
2.
The intensities of diffraction peaks decrease because it has the effect of smearing out the lattice planes.
3.
The intensity of background scattering increases.
Theoretically there is no completely thorough way of calculating thermal parameters – which may be isotropic as well as anisotropic. In practice, average thermal displacements (the
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MSE 421/521 Structural Characterization amplitudes of thermal vibrations) are extremely difficult to calculate. In general the temperature factor it is written as: e − 2 M where M = 2 π
2



µ d
2
2



= 8 π
2
µ
2

 sin
λ
θ 

2
= B

 sin
λ
θ 

2
= B ( 4 d 2 ) where B is the half-height breadth of the peak and
µ s
2 is the mean square displacement of the atoms from their equilibrium positions in a direction perpendicular to the reflecting planes of interplanar spacing d . The inclusion of this factor has the effect of reducing the intensities of higher-angle reflections. In effect, the atomic scattering factors are reduced by thermal vibrations to f n e -M , and since the intensity is proportional to f n
2 , the temperature factor is usually written as e -2 M .
F hkl
=
N
∑ n = 1 f n e
2 π i ( hx n
+ ky n
+ lz n
) e 



− B n

 sin θ
λ


2 


Structure Factor of Centrosymmetric Crystals
If a centre of symmetry is present, for every atom at xyz there is another at x y z , so
F hkl
= n
∑
1 f n
[ cos 2 π (hx + ky + lz) + cos 2 π ( − hx − ky − lz)
]
− lz)
]
+ i n
∑ f n
[ sin 2 π (hx + ky + lz) + sin 2 π ( − hx
1
Since cos(-
φ
) = cos
φ
and sin(-
φ
) = -sin
φ
, we have simply:
− ky
F hkl
= 2
N
∑ f n cos 2 π (hx + ky + lz) n = 1
The imaginary part has vanished; therefore, the structure factor of a centrosymmetric crystals is a real number .
Phase problem not completely eliminated, as F hkl
could be positive or negative .
Friedel's Law
For noncentrosymmetric crystal, structure factor for hkl can be written simply as:
F hkl
= A + iB
The structure factor for h k l
is then:
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F h k l
= A – iB
The observed diffraction intensities for these two planes are proportional to:
| F hkl
| 2 = A 2 + B 2
| and
F h k l
| 2 = A 2 + B 2
The two planes give rise to the same diffraction intensity and so the diffraction pattern has a centre of symmetry!
We cannot tell by inspection of diffraction-pattern intensities alone whether or not a crystal has a centre of symmetry.
Laue Classes
Triclinic 1
1
Monoclinic
Trigonal
2
3 m
3
2 m
Orthorhombic 222 2 mm mmm
Tetragonal 4
4
4 m
422 4 mm
32 3 m
3 m
4 2 m
Hexagonal 6
6
6 m
622 6 mm
6 m 2
4 m
6 m mm mm
Cubic 23 m 3 432
4 3 m m 3 m
Because diffraction always adds a centre of symmetry, we can only distinguish between the 11 centrosymmetric groups ( Laue classes ), shown outlined in bold There are 11 Laue classes , each denoted by the highest point-group symmetry in that class. Diffraction can only distinguish between these 11 Laue classes .
Note that some older texts use the term “ Laue group ” instead of “ Laue class ,” which is not strictly correct according to group theory. The point groups within each Laue class do not constitute groups in the mathematical sense. The error was corrected in the 1969 version of the
International Tables.
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The Multiplicity Factor, p
The relative proportion of planes contributing to the same reflection is the multiplicity factor. It is the number of planes in a form having the same spacing. For example, in a powder sample
(always assuming an infinite number of randomly oriented grains) of a material with a cubic crystal structure, some grains might be oriented such that their (100) planes can contribute to a diffraction peak, while there may be other grains whose (010) or (001) planes are correctly oriented. Since all of these planes have the same spacing, reflections from all of them overlap and form part of the same diffraction peak. There are six such {100} planes, namely, (100),
(010), (001), ( 1 00), (0 1 0), and (00 1 ); therefore, p = 6 for {h00} planes in a cubic system. In the case of {111}, there are eight equivalent planes, namely (111),
( 1 11), (1 1 1), (11 1 ), ( 1 1 1 ), (1 1 1 ), ( 1 1 1 ), and ( 1 1 1); therefore, p = 8. We can now say that it is 8/6 = 4/3 more probable that {111} planes will be correctly oriented for diffraction than
{100} planes.
Lorentz Factor
This factor is in fact made up of three separate parts.
First
, a geometrical factor must be considered which accounts for the fact that crystals can produce non-zero diffracted intensity even for angles not exactly equal to the Bragg angle. If a crystal is oriented exactly at the Bragg angle
θ B
, it will obviously produce the maximum diffracted intensity at this angle; however, if it is rotated very slightly by ∆θ ° away from the Bragg angle, such that the incident angle is θ
1
= θ
B
+
∆θ
and the diffracted angle is
θ 2
=
θ B
–
∆θ
, it is still possible to obtain non-zero diffraction intensity. In fact, diffraction occurs in a cone of directions centred on the Bragg angle.
1 ’ 2 ’
Considering the crystal plane above, with an atom spacing a and total length Na , the difference in path length between rays 1 ’ and 2 ’ is given by:
δ = AD − CB
= a cos θ
2
− a cos θ
1
= a [cos( θ
B
− ∆ θ ) − cos( θ
B
+ ∆ θ )]
Now, assuming ∆θ is very small so that sin ∆θ ≈ ∆θ , this expression can be simplified † to:
δ = 2 a ∆ θ sin θ
B
† Recall the trigonometric identities: cos( a + b ) = cos( a) cos( b) – sin( a) sin( b) cos(-a) = cos(a) sin(-a) = -sin(a)
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The path difference between rays scattered at either end of the plane is N times this quantity.
When such rays are one wavelength out of phase, the diffracted intensity will be zero:
2 Na ∆ θ sin θ
B
= λ or ∆ θ =
λ
Na sin θ 2
B which is the maximum angular rotation away from the Bragg angle over which appreciable energy will be diffracted. Since I max
depends on this range, I max
is proportional to 1/sin θ
B
. The integrated intensity is equal to the product of I max
and the half-maximum breadth of the peak, which is proportional to 1/cos θ . The integrated intensity therefore varies with:


 sin
1
θ
B






1 cos θ
B



=
1 sin 2 θ
B
Second
, for a powder sample in a diffractometer, we must also consider the total number of crystals oriented at or near the Bragg angle for a particular reflection. This number is not constant, even if the crystals are oriented completely randomly.
For a particular reflection hkl , the only planes which can contribute are those with the ends of their normals lying in the band of width r ∆θ . The fraction of these is given by the ratio of the area of the strip to that of the whole sphere. This ratio is:
( r ∆ θ ) 2 π r sin( 90 ° − θ ) ∆ θ cos θ
4 π r 2
=
2
So, the number of crystals favourably oriented for the hkl reflection is proportional to cos θ and so is quite small for reflections in the backward direction (2 θ > 90°):
2 θ ∆ θ cos θ
2
0 0.5
∆θ
45° 0.46
∆θ
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MSE 421/521 Structural Characterization
90° 0.35
∆θ
135° 0.19
∆θ
180° 0
Third
, the detector will see a greater proportion of a diffraction cone when the reflection is in the forward or backward direction than it does near 2 θ = 90°.
2
π
( r
The circumference of a diffraction cone will be given by: sin 2
θ
) and so the proportion observable will be proportional to 1/sin2 θ .
These three effects are combined † into a single factor called the
Lorentz factor
:


 sin
1
2 θ


 cos θ


 sin
1
2 θ



= cos sin 2
θ
2 θ
=
4 sin 2
1
θ cos θ
This in turn is combined with a polarisation factor, ½(1 + cos 2 2 θ ), which arises because the incident beam is unpolarised, to give a combined ‡
Lorentz-polarisation factor
:
† Recall that sin2 θ = 2sin θ cos θ
‡ The constant factor 1/8 is omitted.
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MSE 421/521 Structural Characterization
Lp
1
+ cos 2 2
θ
= sin 2
θ cos θ
The overall effect of Lp is to reduce the intensity of reflections at intermediate angles.
0 20 40 60 80 100 120 140 160 180
Diffraction Angle (2
θ
)
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Applications of X-Ray Diffraction
Phase identification
The most obvious use of XRD is for phase identification
. By obtaining a diffraction pattern from an unknown powder sample, it is possible to match the observed peak positions and relative intensities to those tabulated in standard references in order to find a match. Traditionally, this is done by using the Hanawalt search manual. The strongest three diffraction peaks are used first to find a close match, and then the remaining five strongest peaks are used to determine the match. As an example, consider the XRD pattern shown below. How do we know it is sodium chloride?
Diffraction Angle (2 θ )
The search starts by finding the d -spacings of the eight strongest peaks in the pattern. In this case, as the angular range only covers 10
≤
2
θ ≤
60°, only five peaks are apparent (and one of those is very weak). The step size used to obtain the trace was 0.02° and the radiation used was
Cu K α ( λ = 1.540562 Å). The three strongest peaks occur at 2 θ = 31.70°, 45.44°, 56.48°. These angles are transformed into d -spacings with the Bragg law:
2 d sin
θ = λ
∴ d =



λ
2 sin θ



So, the corresponding d -spacings are: 2.82 Å, 1.99 Å, and 1.63 Å, and their normalised intensities are 100, 63, and 19. The Hanawalt search manual indicates intensities as subscripts to a single digit only, so these peaks would appear as
2.82
x
, 1.99
6
, and 1.63
2
. We now consult the
Hanawalt search manual. The manual is arranged in descending order of d , first by the strongest peak, then the second-strongest peak, and finally by the third strongest. By consulting this manual, we find that there are two possibilities, BaFe
0.24
Fe
0.76
O
2.88
or NaCl, but the intensities are a better fit for NaCl. To be sure, we must look at the other peaks in the pattern. They are
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MSE 421/521 Structural Characterization
27.36° (
3.26
1
) and 53.86° (
1.70
tiny peak at 1.70
0
0
). A good match for the 3.26
1
peak is found with NaCl. The
does not appear in either compound – it is probably too weak to be among the eight strongest reflections in either case. Because the highest angle used was 60°, the smallest possible d -spacing obtainable is 1.54 Å, no spacings smaller than can be detected without extending the scan to higher angles. By a quick process of elimination, we conclude that the powder sample is the mineral halite, NaCl.
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MSE 421/521 Structural Characterization
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MSE 421/521 Structural Characterization
Once this determination is made, we can consult the appropriate JCPDS (Joint Committee for
Powder Diffraction Standards) card. The card number is given on the right hand side of the entry for NaCl in the search manual (5-628). The first number is the volume (5), while the second number is the number of the card within that volume.
This card tells us that this material is cubic, has space group Fm3m , that its lattice constant is a =
5.6402 Å, and that it’s colourless. The star at the top right indicates that this pattern has been rigorously verified and the International Centre for Diffraction Data has very high confidence in its accuracy. We can index
the peaks in our diffraction pattern according to the card, which tells us that the peak at 31.70° (2.82 Å) corresponds to the {200} planes, the peak at 45.44° (1.99 Å) corresponds to {220} planes, etc.
If our powder sample had other crystalline phases present in it as well, these would have produced their own diffraction peaks and made the indexing of the pattern more difficult – but not impossible.
Structure determination
Another use of XRD is crystal structure determination
. As we have already seen an equation which relates the structure of a crystal to its XRD pattern, it is possible to calculate the theoretical pattern of a model structure and compare the result to that obtained experimentally.
The method for doing this is called the Rietveld method, after Hugo M. Rietveld who introduced the method in 1966. By comparing the calculated pattern to that obtained experimentally, and refining the structural model until the two patterns match, it is possible to determine the crystal structure of an unknown substance.
λ = 1.540562 Å
R. Ubic
Diffraction Angle (2 θ )
Rietveld refinement of XRD data for La(Zn
½
Ti
½
)O
3
.
III-28

MSE 421/521 Structural Characterization
Phase diagram construction
XRD can also be used for the determination of phase diagrams
. If a sample is heated (or cooled) inside the diffractometer, then the phases present can be determined by XRD as a function of composition and temperature. It is a also possible, but not as accurate, to use samples which have been heated to some temperature and then quenched back to room temperature before inserting into the diffractometer.
Strain calculation
Strain
can also be detected by XRD. If a material is under tensile strain, the crystals are all slightly enlarged, thus altering interplanar spacings in a way which can be detected by very careful XRD experiments.
Crystallite size determination
As the widths of diffraction peaks are dependent upon crystallite size, an estimate of crystallite size can be made via XRD (note: this is NOT the same as grain size or particle size!). This is true because, although Bragg diffraction is strongest when the Bragg condition is exactly satisfied, its intensity does not become zero for very small deviations away from the Bragg angle. Consider a crystal of thickness x and a set of crystal planes with interplanar spacing d .
For a small deviation,
δθ
, away from the Bragg angle,
θ
, the number of planes which contribute to a particular reflection is x / d .
θ + δθ θ + δθ
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MSE 421/521 Structural Characterization
The path difference between the two rays shown is: †
2 d sin( θ + δθ ) = 2 d (sin θ cos δθ + cos θ sin δθ ) ≈ 2 d sin θ + 2 d cos θδθ
The left-hand term we recognise from the Bragg law. The right-hand term is the extra path length caused by the angle being slightly different from the exact Bragg angle. The phase error
(in radians) is therefore:
2 π
2 d cos θδθ
λ and rays from each successive plane will be this much out of phase with those from the one before. When the total of this phase difference is equal to 2 π , the net contribution of these offangle reflections drops to zero. By multiplying the number of such planes in the crystal by this phase difference, we arrive at the total phase difference between the bottom plane and the top plane. When this difference is equal to 2 π we obtain the first minimum in diffracted intensity.


 x d






2 π



2 d cos θδθ = 2 π
λ
∴ δθ
λ
=
2 x cos
θ
Diffraction angles are usually measured in 2 θ rather than θ , so we can re-write this expression in a more convenient form:
δ
( 2
θ
)
λ
= x cos
θ
A more rigorous treatment of the problem shows that, in fact,
δ ( 2 θ ) = k λ x cos θ where k is a constant about which there has been considerable disagreement. Values given include 0.94 (Scherrer, 1920), 0.89 (Bragg, 1933), 0.92 (Seljalkow, 1925), and 1.42 (Laue, 1926;
Jones, 1938). The value of k depends on the shape of the crystals and, for non-spherical particles, the particular reflection. For spherical particles made of spherical crystals, Stokes and
Wilson gave k = 1.0747.
Since all of these values of k are very near unity, most users of this theory simply neglect k and refer to x as the crystallite size. From the Scherrer equation, one can see that for a given crystal size the broadening becomes most pronounced at high values of θ .
In addition, x-ray line breadth contains information on lattice strain (due to impurities), lattice defects, and thermal vibrations of the crystal structure.
† Recall that sin( a + b ) = sin( a )cos( b ) + sin( b )cos (a )
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MSE 421/521 Structural Characterization
The chief problem to determine crystallite size from line breadth is the determination of δ(2θ)
(usually termed β) from the diffraction profile, since broadening can also be caused by the instrument. To correct for instrumental broadening in the pattern of the sample, it is convenient to run a standard peak from a sample in which the crystal size is large enough to eliminate all crystallite size broadening. The standard peak is run under instrumental conditions identical with those of the sample, so that the broadening of the standard can be equated with the instrumental broadening. By use of a convolution integral method with these two patterns, the crystallite size broadening can be isolated.
This equation is known as the
Scherrer formula
. Because δθ is infinitesimally small, only an infinite number of planes (and therefore an infinitely thick crystal) would yield an infinitely sharp diffraction maximum at exactly the Bragg angle. For any finite thickness, a contribution to intensity will be made even at angles slightly different to θ B
. We can therefore use the breadth of a diffraction peak to calculate the thickness of our crystallites. For example, let’s look again at
NaCl and see what happens to the diffraction pattern as the crystallite size changes.
If the crystallite size is too large, the narrowness of the diffraction peaks and the finite step size of most detectors actually makes detecting diffraction very difficult. For example, the halfheight width of the 200 peak from a 1mm crystallite of NaCl would be only 0.0000093° - too small to be observable.
† Such a crystallite would contain about 3.5 million {200} planar spacings. For crystallites only 500Å thick, only about 177 {200} planar spacings are present and the width of the resultant peak is 0.19°. Crystallites only 100Å thick contain only about 35
{200} planar spacings and the width of the corresponding diffraction peak is a whopping 0.93°. x = 500 Å x = 100 Å
0
10
Diffraction Angle (2 θ )
† Actually, instrumental broadening, typically around 0.15°, usually ensures that even these peaks are detected.
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MSE 421/521 Structural Characterization
Other factors like instrumental broadening, lattice strain, and K α
1
/ K α
2
splitting influence the width of diffraction peaks as well and really should all be taken into account when precisely calculating the crystal size by XRD.
The chief problem in determining crystallite size from XRD line breadth is the determination of the appropriate peak broadening, since broadening can also be caused by the instrument and strain. Assuming that strain can be neglected, then to correct for the instrumental broadening in the pattern it is normal procedure to run a standard peak from a sample in which the crystallite size is large enough to eliminate the crystallite-size broadening effect (LaB
6
is common). The standard is run under instrumental conditions identical to those of the sample, so that the broadening of the standard can be equated with the instrumental broadening. By use of a convolution integral method with these two patterns, the crystallite size broadening can be isolated.
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Laue X-Ray Diffraction
Laue diffraction is a method used to orient single crystals. It is based on the first ever diffraction experiment. A single crystal is placed on a three-axis goniometer in front of a narrow beam of xrays. The x-rays are not monochromatic but the white radiation from an x-ray tube. The Bragg angle and d , of course, remain fixed for each set of planes, and each set picks out and diffracts that particular wavelength which satisfies the Bragg law for it; therefore, each diffracted beam will have a different wavelength. Typically, the beam passes through a slide of photographic film, hits the crystal, and is back-reflected towards the film, where a spot pattern is formed.
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Rotating Crystal Method
In the rotating-crystal method a single crystal is mounted with one of its axes normal to a monochromatic x-ray beam. A cylindrical film is placed around the sample and as the sample rotates, some sets of planes will momentarily satisfy the Bragg condition and form a diffraction spot on the film. When the film is laid out flat, a series of horizontal lines appears. Since the crystal is rotated only about a single axis, the Bragg angle does not take on all possible values for every set of planes; therefore, not every set is able to produce a diffracted spot. This method is sometimes used in the determination of unknown crystal structures.
It is necessary to vary
Method
Laue
λ or θ to satisfy Bragg's Law.
λ
White radiation
(mixed)
θ
Detector (film) does not move
Rotating Crystal Monochromatic
(fixed)
Powder Diffraction Monochromatic
(fixed)
Some variation
(rotating single crystal)
Sample as fine powder with random orientation
Use
Determine the orientation of single crystals.
As above, e.g., silicon, germanium etc.
Most common method of determination.
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Neutron Diffraction
Neutron diffraction was first demonstrated by Von Halban and Preiswerk in 1936. Neutrons are better tools than x-rays for a number of applications, and neutron diffraction is very different from XRD in a number of ways.
1.
Neutrons are highly penetrative. An iron plate 1cm thick is opaque to electrons, virtually opaque to 1.5 Å x-rays, but transmits 35% of 1.5 Å neutrons.
2.
Neutron-scattering powers of atoms vary irregularly with atomic number. Two elements may differ in atomic number by only one unit and yet their neutron scattering powers may be entirely different. Also, some light elements, like carbon or oxygen, scatter neutrons more intensely than some heavy elements, like tungsten. Neutrons can distinguish between elements which scatter x-rays with almost equal intensity and are particularly good at detecting order in alloys and compounds.
3.
Neutrons have a small magnetic moment. If the scattering atom also has a net magnetic moment, the two interact and modify the total scattering. In materials that have an
Monochromatic neutron diffraction
Monochromatic thermal neutrons, siphoned off from a nuclear reactor, have wavelengths given by: ordered arrangement of atomic moments (antiferromagnetic, ferrimagnetic, and ferromagnetic) neutron diffraction can show both the magnitude and direction of the moments.
λ = h
2 mE where h is Plank’s constant ( h = 6.6261 × 10 -34 Js), m is the rest mass of a neutron ( m =
1.67492716 × 10 kT and therefore
-27 kg), and E is kinetic energy. The largest fraction of such thermal neutrons have kinetic energies equal to kT , so when these are selected by the monochromating crystal, E =
λ = h
2 mkT where k is Boltzmann’s constant ( k = 1.3807 × 10 -23 J/K), and T is absolute temperature. As T is generally 300K to 400K, λ is typically 1 – 2 Å (the same order of magnitude as x-ray wavelengths). Instruments like d2b at the Institut Max von Laue – Paul Langevin (ILL) neutron facility in Grenoble operate like this.
Time-of-Flight Method
Another way of using neutrons is with a pulsed source of polychromatic radiation. In such an experiment, wavelengths of the neutrons are determined by their time of flight
(ToF) between the source and the detector. The energy of these neutrons is still: h
λ =
2 mE
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MSE 421/521 Structural Characterization but since not all the neutrons will have the same energy (the beam is not monochromatic), there will be a range of wavelengths present in the beam. In any case, the kinetic energy will be given by E = ½ mv 2 . Making this substitution yields: h h 6 .
62606876 × 10 − 34 Js 3 .
956033981 × 10 − 7 m 2 /s
λ =
2 m 1
2 mv 2
= =
( 1 .
67492716 × 10 − 27 kg ) v
= mv v m 2
=
( 3 .
956033981 × 10 − 7 /s) t
L where L is the path length of the neutron and t is the time of flight. For convenience, we put λ in terms of Å, L in m, and t is µ s. In that case,
λ = 0 .
003956033 ( t / L )
λ L
∴ t =
0 .
003956033 from the Bragg law, we know that λ = 2 d sin θ . Substituting this expression into the above equation yields: t =
2 dL sin θ
0 .
003956033
=
505 .
5568304 Ld sin
θ
So, for a given set of planes ( d ) and angle ( θ ), the ToF will be a linear function of L . If L is large, then ToF is large. The resolution of such an instrument is maximised for long ToF, long path lengths, and large angles ( θ ≈
90°); therefore, the highest accuracy can be achieved by operating with a very long L and in a back-scattered mode. The HRPD instrument at the ISIS facility in the Rutherford Appleton Laboratory operates like this, with the beam source 95m away from the sample!
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Diffraction Angle (2 θ )
λ = 1.540562 Å
Rietveld refinement of neutron diffraction data for La(Zn
½
Ti
½
)O
3
. Note the strong “ordering” peak at d = 4.56 Å which is almost invisible by XRD (19.46° in the inset).
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Electron Diffraction
Geometry of Electron Diffraction
As we have already seen, electrons, although particles, can behave as waves and so can be diffracted. The wavelength of an electron accelerated by an electric field is given by:
λ =
(
2
2
+ e V
) −
1
2
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 , tan2 θ ≈
2 θ . Combining this relationship with Bragg’s law yields:
R L = λ d
∴ d = 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 2 θ even more rapidly than for XRD, which means that the entire observable diffraction pattern is limited to an angular range of about ±4° 2 θ . We will cover this topic in more detail later on.
R. Ubic III-38
[001] electron diffraction pattern from an fcc crystal.

MSE 421/521 Structural Characterization
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 every electron diffraction pattern.
Rd = L λ , which is the equation used to index
L
2 θ
Sample
(d = planar spacing)
2dsin θ = λ
sin
θ ≈ θ tan(2 θ ) = R/L tan(2 θ )
≈
2 θ
∴θ ≈
R/2L
∴
Rd
≈
L
λ
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 angle. The angle 2
θ is the Bragg of the electrons, so Bragg’s Law reduces to 2d
Additionally, it can be seen that tan(2 to 2
θ is very small due to the small wavelength
θ ≈ λ .
θ ) = R/L , which reduces
θ ≈
R/L . Combining these two equations yields Rd
≈
L λ .
T
R
D
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:
F
( hkl )
= n
∑
1 f n e 2 π i ( hu n
+ kv n
+ lw n
) where f is the atomic scattering factor (a function of sin θ / 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 written as: † n th atom are ( u n v n w n
) 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-
F = n
∑ f n
[cos 2 π ( hu n
+ kv n
+ lw n
) + i sin
1
With this definition in mind, it is fairly easy to prove that:
2 π ( hu n
+ kv n
+ lw n
)]
† Recall that e ix = cos x + i sin x
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MSE 421/521 Structural Characterization e π i
= e 3 π i
= e 5 π i
= − 1 e 2 π i e n π i
=
= e 4 π i
= e 6 π i
= + 1 e − n π i where n is any integer
=
4 ( f
Na
+ f
Cl
) if ( h + k + l e n π i
) is even
=
These simple rules enable us to calculate the structure factors for most reflections; however, there are certain special cases in which we are required to use the trigonometric functions instead, e.g.
, if ( hu n
+ kv n
+ lw n
)
≠
0, ½, 1...).
Now let’s look at a real crystal – the simple case of NaCl. This crystal has face-centred symmetry and, because it consists of two different ions, can be described as two interpenetrating fcc lattices offset from each other by (½ 0 0); therefore, there are eight ions per unit cell. The coordinates are:
Na: (0 0 0) (½ ½ 0) (½ 0 ½) (0 ½ ½)
Cl: (½ ½ ½) (½ 0 0) (0 ½ 0) 0 0 ½)
F
F
=
= f
Na e 2 π i f
Na
(
1 +
+ e f
Na e
2
π i
( h
2
+ k
2
)
π i ( h + k )
+ e
+ f
Na
π i ( h + l ) e
2
+
π i e
( h
2
+ l
2
)
π i ( k + l )
+
) f
+
Na e
2 f
π i
Cl
(
( k
2
+ l
2
) e π i ( h
+ f
Cl e
2
π i
( h
2
+ k
2
+ k + l )
+ e π ih
+
+ l
We can factor out an e π i ( h+k+l ) from the Cl term and obtain:
F = f
Na
(
1 + e π i ( h + k )
+ e π i ( h + l )
+ e π i ( k + l ) )
+ f
Cl e π i ( h + k + l ) (
1 + e − π i ( k + l )
2
)
+ e π ik
+ f
Cl e
2
π i
+ e π il h e − π i ( h + l )
)
+
+ f
Cl e
2
π i e k
− π i ( h + k )
+
) f
Cl e
2
π i l
Now, because e n
F = f
Na
(
1 + e
π i = e n
π i , the signs of the exponents in the Cl term can be changed:
π i ( h + k )
+ e π i ( h + l )
+ e π i ( k + l ) )
+ f
Cl e π i ( h + k + l ) (
1 + e π i ( k + l )
+ e π i ( h + l )
+ e π i ( h + k ) )
This expression can be re-grouped:
F =
(
1 + e π i ( h + k )
+ e π i ( h + l )
+ e π i ( k + l ) )( f
Na
+ f
Cl e π i ( h + k + l ) )
Here, the first set of parentheses corresponds to the face centring, whereas the second term corresponds to the basis of the unit cell – the one Na + and one Cl associated with each lattice point. The first part vanishes if h , k , and l are mixed odd and even, otherwise it is equal to 4. For unmixed indices (all odd or all even),
F =
4 ( f
Na
+ f
Cl e π i ( h + k + l ) )
(
−
1 ) n where n is an integer
=
4 ( f
Na
− f
Cl
) if ( h + k + l ) is odd
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MSE 421/521 Structural Characterization
Introducing a second atom type has not eliminated any reflections allowed, but it has decreased the intensity of some. For example, the (111) reflection now involves the difference, rather than the sum, of the scattering powers of the two ions. This result is independent of which way round the Cl and Na + are labelled in the unit cell. It is the relative position of the atoms which is important.
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 regular diffraction pattern.
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MSE 421/521 Structural Characterization
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 a o
, b o
, c o
, then the reciprocal lattice’s dimensions are a * = 1/ a o
, b * = 1/ b o
, c * = 1/ c o
; and a * is normal to b o
and c o
, b * is normal
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MSE 421/521 Structural Characterization to a o
and c o
, and c * is normal to a o
and b o
. 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).
[001]
[111]
(202)
(002)
(111)
(222)
(022)
The reciprocal of an fcc real lattice is bcc.
[100]
[110]
[010]
(200)
000
(220)
(020) a fcc
1/a bcc
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 K α 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 72,000 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.
Ewald Sphere
1/ λ
Ewald sphere construction.
Incident
Beam
1/ λ
SOLZ
FOLZ
Reciprocal
Lattice Planes
ZOLZ
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MSE 421/521 Structural Characterization
Diffraction Theory
There are two sets of theory and equations which explain electron diffraction; kinematic theory and dynamical theory. The kinematic
theory is simpler and relies upon a few approximations and assumptions. First, the intensity of diffracted spots must be much smaller than the intensity of the transmitted beam. Second, it assumes that electrons scatter only once in the specimen and do not subsequently interact in any way with each other or undiffracted electrons. This is approximately the case for exceptionally thin specimens or when the geometry is far from the
Bragg angle. The dynamical
theory, on the other hand, accounts for absorption by thick specimens and the multiple interactions of electrons which scatter and can be re-scattered. The intensity of the diffracted waves can be almost as large as that of the transmitted waves.
Dynamical scattering makes it much more difficult to calculate the intensity of diffracted spots, as events like double diffraction can redistribute diffracted intensity from one spot to another - even into forbidden reflections.
Double Diffraction
Double diffraction is a dynamical effect whereby electrons which have been Bragg diffracted by one set of planes now satisfy the Bragg condition for another set of planes, and so are diffracted twice. The figure below shows two DP’s, (a) has been calculated kinematically, and (b) is an actual experimental DP obtained from a JEOL 200CX (200 kV). The difference in intensities is obvious, and a few reflections which occur in (b) are not predicted by (a). For any doublediffracted spot, there must be a diffraction path which allows it. For example, the (002) spot which occurs in (b) is forbidden by the extinction rules and so does not appear in (a); however, since both ( 1 1 1 ) and ( 1 11 ) are allowed, and ( 1 1 1 ) + ( 1 11 ) =(002), there is a legitimate route for double diffraction, and the (002) spot appears. A short, visual way of determining if a double diffraction route exists is to overlay the simulation (tracing it on acetate helps) on the experimental DP and aligning the simulated transmitted spot onto each real diffraction spot. If the simulation contains a diffracted spot which coincides with an extra spot on the experimental
DP, then there is a route for double diffraction. Essentially, this is the same as turning each diffracted beam into another transmitted beam, which is exactly what double diffraction is.
006 442
111
220 111
111 220
111
002 111 220
111 111
220 111 002
442 006 a b
(a) Kinematic simulation and (b) experimental DP of fcc Nd
2
Hf with the beam parallel to [110] (zone axis = [110]).
2
O
7
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MSE 421/521 Structural Characterization
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.
β
2 θ
2 θ
α
E g D T
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.
λ L/d sLd s > 0 s = 0 (Bragg) s < 0
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 (for s
≥
0) 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
R. Ubic III-45

MSE 421/521 Structural Characterization 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 passing through the reciprocal lattice point. The rods, called rel rods , are thin in x and y but thick in z .
They are also longer for thinner crystals. The fact that reciprocal lattice points are, in fact, threedimensional rods and not mathematically defined zero-dimensional points makes it possible to get diffracted intensity even when the Bragg condition is not exactly satisfied. The deviation parameter
, s
, is defined as the deviation from the exact Bragg condition where the Ewald sphere cuts the rel rod.
Indexing Diffraction Patterns
The method of indexing diffraction patterns (labelling the spots with appropriate hkl ’s) varies according to what is known of the specimen (crystallography) and diffraction conditions
R. Ubic III-46

MSE 421/521 Structural Characterization
(camera length and voltage). Typically, an experimentalist will have a fairly good idea of what phase or possible phases produced the pattern and a reasonable knowledge of L and λ (the product L λ is often referred to as the camera constant
). In such a case, the work is much simplified; however, if the crystallography is completely unknown, the procedure is complex and requires a lot of trial and error.
To begin indexing a diffraction pattern (DP) such as the one below, a list of R spacings is required. A good idea is to put a ruler across the negative and measure the space across an entire systematic row of spots, and take an average for the space between each spot. The corresponding d -spacings can then be calculated as L λ /R . These d -spacings can be checked against a list of d -spacings for various possible phases. A good start is the JCPDS card files, which usually contain sufficient crystallographic information to calculate d -spacings. Since more reflections are possible in electron diffraction by double diffraction
than occur by XRD, the JCPDS cards may not list all the d -spacings observed in the DP, and the remaining spacings must be calculated. There are a few simple equations for calculating d -spacings of planes in various crystal systems. In the case of a cubic crystal,
1 h
2
+ k
2
+ l
2 d
2
= a
2
Once generic hkl ’s are found, they must be made systematic.
In the case of the DP below, | g
1
| = | g
3
| and both correspond to a d -spacing of 6.14 Å, matching the {111} of fcc Nd
2
Hf
2
O
7
. The magnitude of vector g
2
corresponds to d = 3.76 Å, matching
{220} of Nd
2
Hf
2
O
7
. From the DP, it is clear that g
1
+ g
3
= g
2
, and this must be the case for the hkl ’s. With this rule, the spots are made systematic by indexing g
1
as ( ) g
2
as ( 220 , and g
3
as ( ) g
1
+ g
3
= ( ) ) 220 = g
2
. Every other spot in the pattern is now determined by similar vector addition.
006 442 Diffraction pattern from fcc Nd
2
Hf
2
O
7
.
111
220
002 111 g1
220 g2
111 g3
111 002
442 006
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
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MSE 421/521 Structural Characterization correct. For cubic crystals, the angle h
2 k
2 l
2
, is given by:
φ between two planes (two plane normals), h
1 k
1 l
1
and cos
φ = h h +
1 2
+
1 2
( h
1
2
+ k
1
2
+ l
1
2 )( h
2
2
+ k
2
2
+ l
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).
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MSE 421/521 Structural Characterization a
[ mnp ] h1 k1 l1 h1 k1 l1 h2 k2 l2 h2 k2 l2 b
(a) Generic illustration of a zone axis.
(b) Determination of the zone axis of the diffraction pattern in the figure above.
1+1,1+1,-1+1 = [ 220 ] = [110]
Of course, this is exactly equivalent to taking the cross product of the two g
vectors: i j k det 1
1
1
1
1
1
= i ( 1
+
1 )
− j (
−
1
−
1 )
+ k (
−
1
+
1 )
=
[ 220 ]
=
[ 110 ]
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: g • r = 0
( hkl ) • [ mnp ] = 0 hkl × mnp cos
θ =
0
∴ cos
θ =
0
∴ θ =
90
°
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MSE 421/521 Structural Characterization
Convergent Beam Electron Microscopy (CBED)
CBED is a powerful tool for probing the crystallography of a small specimen as well for analysing local strains and thicknesses. Conventional diffraction is performed using nearlyparallel electron beams (parallel illumination), and so gives rise to diffraction spots . The CBED technique involves focusing the incident illumination down to a fine point, and so yields diffraction discs. Because of the extreme intensity of the focused incident beam, the specimen used must be very stable against the radiation. The specimen will become very hot in the region of the fine probe, and so cooling is generally required to prevent thermal diffuse scattering from obscuring the fine detail within the discs. For the same reason, a very high vacuum system is required to prevent dirt (mostly carbon) from being deposited on the specimen surface.
Additionally, the tiny region examined must be defect-free and, if crystallographic information is required, of uniform thickness. Clearly, CBED is not a trivial operation, and the interpretation of
CBED patterns is no less difficult! From CBED images it is possible to obtain local specimen thickness (useful when, e.g.
, determining defect densities), lattice parameters, and crystal structure (even a full space group determination).
Incident
Electrons
Specimen
Diffraction Discs
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